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Question 282707: The area of a square is 12 more than its perimeter. Find the length of a side.
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.
.
The area of a square is 12 more than its perimeter. Find the length of a side.
~~~~~~~~~~~~~~~~~~~~

The solution in the post by @mananth is incorrect, since his setup equation is incorrect.
I came to bring a correct solution.

Let x be the side length of the square.


The setup equation is

    x^2 = 4x + 12,

according to the problem.


Reduce it to the standard form of a quadratic equation and factorize

    x^2 - 4x - 12 = 0,

    (x-6)*(x+2) = 0.


The roots are x= 6  and  x= -2.


Since we look for a side length, we choose the positive root and reject the negative one.


ANSWER.  The side of the square is 6 units.

Solved correctly.

You may check mentally that the answer value satisfies the problem's condition.

Question 1210560: Use a ruler to measure the dimensions to the nearest 1/16th.
Individually, properly dimension the multiview drawing below. Calculate the number of dimension lines you will need. Add the missing lines and draw the missing view. Use your ruler and ignore the graph paper.
https://ibb.co/PKjt9XX
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.
Use a ruler to measure the dimensions to the nearest 1/16th.
Individually, properly dimension the multiview drawing below. Calculate the number
of dimension lines you will need. Add the missing lines and draw the missing view. Use your ruler and ignore the graph paper.
https://ibb.co/PKjt9XX
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The goal and the profile of this forum is to help school/college students
solving their Math problems.

It is not our goal and/or our profile to measure the dimensions for visitors
using the rulers or to make plots, since we are not a graphing studio.

Therefore, to come to us with requests similar to that in your post
is the same as to come to a grocery store for hardware - it has the same meaning,
or, better to say, does not have meaning at all.

Question 1210561: Use a ruler to measure the dimensions to the nearest 1/16th. Individually, properly dimension the multiview drawing below. Calculate the number of dimension lines you will need. Add the missing lines and draw the missing view.
https://ibb.co/N2KmvtLD
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.
Use a ruler to measure the dimensions to the nearest 1/16th. Individually, properly dimension
the multiview drawing below. Calculate the number of dimension lines you will need. Add the missing lines and draw the missing view.
https://ibb.co/N2KmvtLD
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The goal and the profile of this forum is to help school/college students
solving their Math problems.

It is not our goal and/or our profile to measure the dimensions for visitors
using the rulers or to make plots, since we are not a graphing studio.

Therefore, to come to us with requests similar to that in your post
is the same as to come to a grocery store for hardware - it has the same meaning,
or, better to say, does not have meaning at all.

Question 1210562: Use a ruler to measure the dimensions to the nearest 1/16th. Individually, properly dimension the multiview drawing below. Calculate the number of dimension lines you will need. Add the missing lines and draw the missing view.
https://ibb.co/XxjJD23s
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.
Use a ruler to measure the dimensions to the nearest 1/16th.
Individually, properly dimension the multiview drawing below. Calculate the number of dimension lines you will need.
Add the missing lines and draw the missing view.
https://ibb.co/XxjJD23s
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The goal and the profile of this forum is to help school/college students
solving their Math problems.

It is not our goal and/or our profile to measure the dimensions for visitors
using the rulers or to make plots, since we are not a graphing studio.

Therefore, to come to us with requests similar to that in your post
is the same as to come to a grocery store for hardware - it has the same meaning,
or, better to say, does not have meaning at all.

Question 1201550: Not counting rotations and reflections, there are only two different shapes that can be made by joining three squares together. Again not counting rotations and reflections, the number of different shapes that can be made by joining four squares together is
a) 3 b) 4 c) 5 d) 6 e) 7
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.
Not counting rotations and reflections, there are only two different shapes that can be made by joining three squares
together. Again not counting rotations and reflections, the number of different shapes that can be made by joining four
squares together is
a) 3 b) 4 c) 5 d) 6 e) 7
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The post by @mananth has no any relation to the posed question.

Without additional explanations, the meaning of this assignment is dark and unclear.

For complete explanation, see, for example, this site

https://www.mathsisfun.com/geometry/tetromino.html

Question 446858: Find the slope of AC and BD. Decide whether AC is perpendicular to BD.
A(-1,-2) B(-3,2) C(0,1) D(3,0)
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.
Find the slope of AC and BD. Decide whether AC is perpendicular to BD.
A(-1,-2) B(-3,2) C(0,1) D(3,0)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is presented in monstrous form.
        He correctly determined that m1 = 1/3, but then he writes m2 = -0.33.
        This is wrong. What is TRUE, is that   m2 = -1/3, but not -0.33.
        So, the correct presentation should be as showed below.

Slope of Line Ac
x1 y1 x2 y2
-1 -2 0 1
slope m1 =(y2-y1)/(x2-x1)
(1-(-2)/(0-1)
(3/1)
m1 = 3

Slope of line BD
x1 y1 x2 y2
-3 2 3 0
slope m2 =(y2-y1)/(x2-x1)
(0-2)/( 3-(-3) )
(-2/6)
m2 = -1/3
m1*m2=-1
so the lines are perpendicular

This is the correct form to present the solution.

----------------------

For @mananth, -1/3 and -0.33 is the same value, but it is not true - they are DIFFERENT !

In Math, to teach that -0.33 is the same as -1/3 - this is a CRIME ( ! )

Question 203940: I need your help about this:
Find the equation of the line with points equidistant from (5,-2) and (4,3).
Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
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Find the equation of the line with points equidistant from (5,-2) and (4,3).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As an alternative to the geometric solution by @Theo, below is simple algebraic solution.

Let (x,y) be the point equidistant from these given points (5,-2) and (4,3).


It means that the distances from (x,y) to these points are equal.

Hence, the squares of distances are equal.


So, we write

     +  =  + .

or

     +  =  + .


Now we simplify it

    x^2 - 10x + 25 + y^2 + 4y + 4 = x^2 - 8x + 16 + y^2 - 6y + 9,

    -10x + 4y + 29 = -8x - 6y + 25,

    (-10x + 8x) + (4y + 6y) = 25 - 29,

    -2x + 10y = -4,

     -x +  5y = -2,

      x -  5y = 2.

Solved.

Question 554469: hi, i need your help please... i don't know how to answer these questions because i can't fully imagine them... i hope you can help me because i really want to understand this lesson for the sake of my grades.....
a) the sum of the distance from a point P to (4,0) and (-4,0) is 9. if the abscissa of P is 1, find its ordinate...
b)the center of a circle is at (-3,-2). if a chord of length 4 is bisected at (3,1), find the length of the radius...

THANK YOU!!!!!
Found 3 solutions by n2, greenestamps, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
(a) the sum of the distance from a point P to (4,0) and (-4,0) is 9. if the abscissa of P is 1, find its ordinate.
(b) the center of a circle is at (-3,-2). if a chord of length 4 is bisected at (3,1), find the length of the radius.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        I am @ikleyn. Hello again.
        This 'n2' is my second nickname, which I created to place here my solution to part (a) of the problem.

        Tutor @greenestamps solved this part nicely by applying theory of ellipses.
        In my solution below, I work in the frame of knowledge related to triangle geometry, only.

Let A, C and P be the given points A = (-4,0), C = (4,0). Let P = (1,y) be the point, for which we want to find its coordinate 'y' such that |AP| + |CP| = 9. (1) Notice that line AC is horizontal (lies on x-axis of the (x,y) coordinate system). Draw a perpendicular PD from vertex P of triangle APC to its base AC, so point D = (1,0) is the intersection of the perpendicular with AC. Thus we have now triangle APC and two right-angled triangles ADP and CDP. Let 'a' be the length CP and 'c' be the length AP: a = |CP|, c = |AP|. (2) We have |AD| = 1 - (-4) = 5; |CD| = 4 - 1 = 3. (3) Perpendicular PD is the common leg of triangles ADP and CDP, so we can write, using Pythagorean equation |AP|^2 - |AD|^2 = |CP|^2 - |CD|^2. Substituting here from relations (2) and (3), we get this equation c^2 - 5^2 = a^2 - 3^2, which implies c^2 - a^2 = 5^2 - 3^2, c^2 - a^2 = 16. (4) So, now we have this system of equations (1) and (4) c + a = 9. (1') c^2 - a^2 = 16, (4') Now we are on a finish line to complete the solution. In equation (4'), factor left side as c^2 - a^2 = (c+a)*(c-a) and replace (c+a) by 9, based on equation (1'). Then instead the system (1'), (4') you will get the system a + c = 9. (1'') 9(c-a) = 16. (4'') Open parentheses in (4'') and multiply equation (1'') by 9 (both sides) 9c + 9a = 81. 9c - 9a = 16, Add the two last equations and get 18c = 81 + 16 = 97, c = 97/18. Subtract the two last equations and get 18a = 81 - 16 = 65, c = 65/18. Now we can find 'y' y^2 = |PD|^2 = |CP|^2 - |AD|^2 = a^2 - 3^2 = = = , y = = 2.010005835, or y = 2.01, approximately. ANSWER. y = = 2.010005835, or y = 2.01, approximately.

Solved.

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Here is an alternative solution to the first problem which yields nearly the same answer that the other tutor got. However the solution from the other tutor shows only one of two answers to the question.

The information that the sum of the distances from the two fixed points (-4,0) and (4,0) is 9 is the classical definition of an ellipse with center (0,0) and the two foci at those two points.

The equation of an ellipse with center (0,0), horizontal semi-major axis a and vertical semi-minor axis b is

The distance from the center of the ellipse to each focus is c, where

So in this problem we know c is 4.

With the sum of the distances from the two foci being 9, we know that a, the length of the semi-major axis, is 9/2.

Since we know a and c, we can determine b.

Now we know the equation of the ellipse:

Now substitute the given abscissa x=1 in the equation to find the ordinate y.

That value to several decimal places is the answer the other tutor got: y = 2.010005835.

But this is an ellipse -- there are two values of y when x is 1.

ANSWERS: 2.010005835 and -2.010005835

---------------------------------------------------------------------

NOTE: For the second problem, see the easy and clear solution from tutor @ikleyn.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
hi, i need your help please... i don't know how to answer these questions because i can't fully imagine them...
i hope you can help me because i really want to understand this lesson for the sake of my grades.....
(a) the sum of the distance from a point P to (4,0) and (-4,0) is 9. if the abscissa of P is 1, find its ordinate.
(b) the center of a circle is at (-3,-2). if a chord of length 4 is bisected at (3,1), find the length of the radius.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        In the post by @Theo, problem (b) is solved incorrectly and his answer is wrong.
        I came to bring a correct solution.

Let P = (-3,-2) be the center of the circle. Let C = (3,1) be the midpoint of the chord. Let point A be one of the intersections of the chord with the circle. Triangle PCA is a right-angled triangle with right angle at vertex C. The leg CA has the length of 4/2 = 2 units (half the length of the chord). The leg PC has the length = = = . Therefore the length of the radius, which is the hypotenuse PA of this right-angled triangle PCA, is r = |PA| = = = = 7. ANSWER. The radius is 7 units long.

Solved correctly.
The solution is short, straightforward and elegant - as it is should be for a Math problem.

-------------------------------

This problem is nice. As I see, it is from a classic source and created by really qualified Math composer.

All input data is carefully selected and polished, so that the output is a round number.
This reveals a professional composer and a good source.

Question 1186939: Suppose that O=(0,0), A=(3.4,0), and B=(2.81,1.92). The arc between points B and C is 12.92 units long.
A. What is the value of (theta)1, the radian measure of Angle AOB?
B. What is the value of (theta)2, the radian measure of Angle BOC?
C. What are the x- and y-coordinates of point C?
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Strictly saying, this problem is posed INCORRECTLY,

since it does not define that point 'C' lies in the same circle as points 'A' and 'B'.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Here's the solution, broken down step by step:
**A. θ₁ (Angle AOB):**
1. **Find the lengths of OA and OB:**
* OA = √((3.4 - 0)² + (0 - 0)²) = 3.4
* OB = √((2.81 - 0)² + (1.92 - 0)²) = √(7.8961 + 3.6864) = √11.5825 ≈ 3.403
2. **Use the dot product formula:**
* OA • OB = |OA| * |OB| * cos(θ₁)
* (3.4 * 2.81) + (0 * 1.92) = 3.4 * 3.403 * cos(θ₁)
* 9.554 = 11.5702 * cos(θ₁)
* cos(θ₁) = 9.554 / 11.5702 ≈ 0.8257
3. **Solve for θ₁:**
* θ₁ = arccos(0.8257) ≈ 0.5994 radians
**B. θ₂ (Angle BOC):**
1. **Find the radius of the circle:** Since OA and OB are radii, and we are working with a circular arc BC, OA is the radius. r = 3.4
2. **Use the arc length formula:**
* Arc length (s) = r * θ₂
* 12.92 = 3.4 * θ₂
* θ₂ = 12.92 / 3.4 ≈ 3.8 radians
**C. Coordinates of Point C:**
1. **Find the angle from the x-axis to OC:** This is θ₁ + θ₂ ≈ 0.5994 + 3.8 = 4.3994 radians.
2. **Use the coordinates formula:**
* x = r * cos(θ₁ + θ₂) = 3.4 * cos(4.3994) ≈ -1.0468
* y = r * sin(θ₁ + θ₂) = 3.4 * sin(4.3994) ≈ -3.2348
**Therefore:**
* θ₁ (Angle AOB) ≈ 0.5994 radians
* θ₂ (Angle BOC) ≈ 3.8 radians
* Coordinates of point C ≈ (-1.0468, -3.2348)

Question 1166974: P is the centre of a circle that passes through O, and O is the centre of a circle that passes through P. If angle C=66 degrees, then the measure of angle OPB, in degrees is:
A) 120
B) 125
C) 116
D) 128
E) 124
https://imageshack.com/i/pnM6OEIVj
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

See my (tutor @ikleyn) solution to this problem under this link

https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.1197237.html

https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.1197237.html

at this forum.

Question 1180852: TV weather forecasters use satellite and radar data to predict where storms will move in order to help viewers know what weather to expect. The map below shows a storm off the eastern coast of the United States. The arrows show the path the heart of the storm traveled over the last 48 hours. If you were a forecaster in the northeast, use the map to answer the following questions.
https://api.agilixbuzz.com/Resz/~0.6NT8GFszQyEepNgJ.U8Pc_1Jw6w7bKhdBO1lgK6sThuzD160XdxwpKkVzzWg/48780464,FA7,11,3/Assets/Media/Images/41.2-HOT3-Weather.jpg
a. What would you tell your Northeast coast audience? Which type of reasoning—inductive or deductive—did you use? Explain.
b. Write an if-then statement to describe your conjecture.
c. Write the inverse of the statement.
d. Write the converse and contrapositive of the statement.
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Here's an analysis of the storm's path and how a forecaster might present the information:
**a. What to tell the Northeast coast audience:**
"Good evening, everyone. As you can see on the map, a storm is currently located off the East Coast. Over the past 48 hours, it has followed a generally westward track, as indicated by these arrows. Based on this observed pattern, we can *induce* that the storm is likely to continue moving in a westerly direction. This means it *could* potentially impact the Northeast coast in the coming days. However, it's important to note that storm tracks can change, so we'll continue to monitor the situation closely and provide updates as they become available. We urge everyone in coastal areas to stay informed and prepare for possible severe weather."
* **Type of reasoning:** Inductive reasoning was used. Inductive reasoning involves making generalizations based on observed patterns or specific instances. The forecaster observed the storm's past track and *induced* that it would likely continue on a similar path. It's important to emphasize the uncertainty inherent in inductive reasoning ("could," "possibly") as the observed pattern doesn't *guarantee* future behavior.
**b. If-then statement (Conjecture):**
If a storm has moved westward for the past 48 hours, then it will continue to move westward.
**c. Inverse of the statement:**
If a storm has *not* moved westward for the past 48 hours, then it will *not* continue to move westward.
**d. Converse and Contrapositive:**
* **Converse:** If a storm continues to move westward, then it has moved westward for the past 48 hours.
* **Contrapositive:** If a storm does *not* continue to move westward, then it has *not* moved westward for the past 48 hours.

Question 1209705: Find the slope of the line
y
=
3
2
x
+ 2
.
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

I'll provide a similar example.

Let's say the example equation is , in which you can write as y = (7/5)x+39

Compare that to y = mx+b
m = 7/5 = slope
b = 39 = y intercept

Use that example to determine the slope of y = (3/2)x+2

Question 1188287: Given triangle ABC with D the midpoint of side AB and E on side BC, F on side AC and G is the intersection of side DE and side BF. BE:EC = 2:3 and DG:GE = 5:8. What is the ratio of BG:GF?
Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website!
Certainly, let's find the ratio of BG:GF.
**1. Utilize Ceva's Theorem:**
* Ceva's Theorem states that for any triangle ABC, if lines AD, BE, and CF intersect at a single point (in this case, point G), then:
(AF/FC) * (BD/DA) * (CE/EB) = 1
**2. Apply the given ratios:**
* BD/DA = 1/1 (since D is the midpoint of AB)
* BE/EC = 2/3
**3. Calculate AF/FC:**
* Using Ceva's Theorem:
(AF/FC) * (1/1) * (3/2) = 1
AF/FC = 2/3
**4. Use Menelaus' Theorem:**
* Menelaus' Theorem states that for any transversal line (in this case, line DE) that intersects the sides of a triangle (triangle ABC) at points D, E, and F, then:
(AD/DB) * (BE/EC) * (CF/FA) = 1
**5. Apply the known ratios and solve for BG/GF:**
* (1/1) * (2/3) * (3/2) * (BG/GF) = 1
BG/GF = 1
**Therefore, the ratio of BG:GF is 1:1.**
Let me know if you have any other questions or problems to solve!

Question 1195764:
Let’s assume the following statements are true: Historically, 75% of the five-star football recruits in the nation go to universities in the three most competitive athletic conferences. Historically, five-star recruits get full football scholarships 93% of the time, regardless of which conference they go to. If this pattern holds true for this year’s recruiting class, answer the following:
a. Based on these numbers, what is the probability that a randomly selected five-star recruit who chooses one of the best three conferences will be offered a full football scholarship?
b. What are the odds a randomly selected five-star recruit will not select a university from one of the three best conferences? Explain.
c. Explain whether these are independent or dependent events. Are they Inclusive or exclusive? Explain.

Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website!
**a) Probability of a full scholarship for a recruit in a top conference:**
* Given:
* 75% of five-star recruits go to top three conferences.
* 93% of five-star recruits get full scholarships.
* Since the probability of getting a full scholarship is independent of the conference chosen (given in the problem), the probability of a five-star recruit in a top three conference getting a full scholarship is simply **93%**.
**b) Odds of a recruit NOT selecting a top three conference:**
* Probability of selecting a top three conference: 75%
* Probability of NOT selecting a top three conference: 100% - 75% = 25%
* Odds are typically expressed as a ratio of the probability of an event occurring to the probability of the event not occurring.
* Odds against selecting a top three conference:
* (Probability of NOT selecting) / (Probability of selecting)
* = 25% / 75%
* = 1/3
* Therefore, the odds of a randomly selected five-star recruit NOT selecting a university from one of the three best conferences are **1 to 3**.
**c) Independence and Inclusivity/Exclusivity:**
* **Independence:**
* The probability of a recruit getting a full scholarship is independent of the conference they choose. This is stated in the problem.
* **Inclusivity/Exclusivity:**
* The events "selecting a top three conference" and "not selecting a top three conference" are **mutually exclusive**. This means a recruit cannot simultaneously select a top three conference and not select a top three conference.
**In Summary:**
* a) Probability of a full scholarship for a recruit in a top conference: 93%
* b) Odds of a recruit NOT selecting a top three conference: 1 to 3
* c) Scholarship probability and conference choice are independent events.
* Selecting a top three conference and not selecting a top three conference are mutually exclusive events.

Question 1208981: One square is placed upon another so that a regular octagon is formed, along with eight right isosceles triangles each with its hypotenuse on a side of the octagon. If the perimeter of the octagon is 18 cm, find the perimeter of the star, in cm.
https://ibb.co/3MVhbWy
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

a TWIN problem was solved at this forum many years ago under this link

https://www.algebra.com/algebra/homework/Length-and-distance/Length-and-distance.faq.question.1148864.html

Question 1208847: P(-2,1) y=1/4x-3
Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

You didn't state what you want. Do you want the equation of the line through P(-2,1) parallel (or maybe perpendicular) to the line y=1/4x-3? Do you want the distance from the point P(-2,1) to the line y=1/4x-3? You must tell us what you want us to do. Otherwise, we can't know. Edwin

Question 1207455: ABC is a triangle with ∠CAB=15
∘
and ∠ABC=30
∘
. If M is the midpoint of AB, Sin then ∠ACM= ?
Found 3 solutions by ikleyn, Edwin McCravy, greenestamps:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
ABC is a triangle with ∠CAB=15∘ and ∠ABC=30∘.
If M is the midpoint of AB, find ∠ACM.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution by Edwin is perfect and deserves admiration.

        My congratulations to Edwin with this great achievement.

        I came to bring a shorter solution.

The given part is shown in Figure 1 below. Figure 1. Triangle ABC has the angles A= 15°, B= 30° and C= 180°-15°-30°= 135°. We want to find angle . Draw perpendicular CH from vertex C to the base AB (Figure 2). Figure 2. First part of the solution is the same as that of Edwin' solution, and it shows that a = , b = . The rest of the solution is different (very simple and totally geometrical). Triangle BHC is a right-angled triangle with the acute angle B of 30°. Hence, the opposite leg CH is half of the hypotenuse BC CH = = , (1) while its other leg BH is times the hypotenuse BC BH = = . Then the segment MH is the complement of BH to 1 MH = 1 - = = . (2) Comparing expressions (1) and (2), we see that CH = MH. Hence, right-angled triangle MHC is isosceles right-angled triangle, which implies that angle MCH is 45°. Thus ∠ACM = 135° - 60° - 45° = 30°, and the problem is solved completely.

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!
I just ignored "Sin".

This problem would be much easier if you could use decimals, but decimals are approximate. The answer could only be approximate if we use decimals, so we must use exact values. Maybe some other tutor can come up with a shorter solution, but this was the very long solution I came up with: We want to find angle θ Without loss of generality we can let AM = MB = 1 unit. We are going to need expressions for both sine and cosine of 15^o. Angle ACB = 180^o-15^o-30^o = 135^o. By the law of sines on triangle ABC. By the law of cosines on triangle ACM Now let's see if we can rewrite that so it doesn't have a square root under a square root. We'll see if we can write it as the difference of 2 square roots: => => => CM can't be negative, so Use the law of sines on triangle ACM: Edwin

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

What does this mean: "... Sin then ∠ACM= ?"

Re-post, making it clear what the question is.

Question 1206891: Indicate in standard form the equation of the line passing through the given points, writing the answer in the equation box below.
G (4, 6), H (1, 5)

Found 2 solutions by ikleyn, josgarithmetic:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Indicate in standard form the equation of the line passing through the given points,
writing the answer in the equation box below.
G (4, 6), H (1, 5)
~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @josgarithmetic is WRONG.
        It is incorrect from the first step calculating the slope and to the end, including his answer .
        I came to bring a correct solution.

Calculate the slope m = = = = . It is the same as to use the formula m = with coordinates = 4, = 6, = 1, = 5. Next, write an equation of the line having the slope and passing through the given point (4,6). An equation of a straight line in a coordinate plane which has the slope m and passes through the given point P = (a,b) is y - b = m*(x-a). Substitute here m = , a = 4, b = 6, and you will get y - 6 = . (1) It is the equation in the slope-point form. But you want to get an equation in standard form, which is Ax + By = C. To get it, multiply equation (1) by 3 (both sides) and simplify 3*(y-6) = x-4, -18 + 4 = x - 3y, -14 = x - 3y, x - 3y = -14. This equation is what you want, i.e. your ANSWER.

----------------

See the lesson
    - Equation for a straight line in a coordinate plane passing through two given points
in this site. Find there several other similar solved problems.

Learn the subject from there.

As a bonus to you, consider the lessons in this site
    - Find the slope of a straight line in a coordinate plane passing through two given points
    - Equation for a straight line having a given slope and passing through a given point
    - Solving problems related to the slope of a straight line
    - Equation for a straight line in a coordinate plane passing through two given points
    - Equation for a straight line parallel to a given line and passing through a given point
    - Equation for a straight line perpendicular to a given line and passing through a given point
that are closely related to your problem.

Consider them as your textbook, handbook, tutorials and (free of charge) home teacher.

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Find slope. Write an equation for slope using either your point G or H, AND any unknown point (x,y). Adjust the form of your equation.

SLOPE:

slope should be .

***************************
Choose points H and (x,y) for writing slope equation again.
H is at (1,5).

---removed---

Question 1206892: Indicate in standard form the equation of the line passing through the given points, writing the answer in the equation box below.
E ( - 2, 2), F (5, 1)

Found 3 solutions by josgarithmetic, Theo, MathLover1:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Through (-2,2) and (5,1) as ax+by=c

Point-Slope form

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
the points are (-2,2), (5,1)
let x1,y1 = -2,2 and x2,y2 = 5,1
slope = (y2-y1)/(x2-x1) = (1-2)/(5+2) = -1/7
use one of the points to get the y-intercept.
general slope intercept form equation is y = mx + b
m is the slope
b is the y-intercept
when the slope is -1/7, equation becomes y = -1/7 * x + b
use (5,1) coordinate and replace y with 1 and x with 5 to get:
1 = -1/7 * 5 + b
solve for b to get:
b = 1 + 5/7 = 12/7
equation becomes y = -1/7 * x + 12/7.
that's the slope intercept form of the equation.
to get the standard form, do the following:
add -1/7 * x to both sides of the equation to get 1/7 * x + y = 12/7
multiply both sides of the equation by 7 to get x + 7y = 12
that's the standard form of the equation.

here's the graph.

note that the standard form and the slope intercept form of the equation draw the same line.
that's because they're equivalent.

the two speciied points are on the line.
they are (-2,2) and (5,1)
the slope intercept is (0,1.714) where 1.714 represents the fraction 12/7.

Answer by MathLover1(20855)   (Show Source):
You can put this solution on YOUR website!

Solved by pluggable solver: Finding the Equation of a Line

First lets find the slope through the points (,) and (,)

Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,))

Plug in ,,, (these are the coordinates of given points)

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

So the slope is

------------------------------------------------

Now let's use the point-slope formula to find the equation of the line:

------Point-Slope Formula------
where is the slope, and (,) is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)

Rewrite as

Distribute

Multiply and to get

Add to both sides to isolate y

Combine like terms and to get (note: if you need help with combining fractions, check out this solver)

------------------------------------------------------------------------------------------------------------

Answer:

So the equation of the line which goes through the points (,) and (,) is:

The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is

Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver )

Graph of through the points (,) and (,)

Notice how the two points lie on the line. This graphically verifies our answer.

....write in standard form , both sides multiply by

Question 1064005: ΔABC has vertices at A(5,2), B(−3,1), and C(−2,5). Point D is located on the intersection of the altitude and AB�, in such a way that D has coordinates at approximately (−1.52,1.18).
https://cds.flipswitch.com/tools/asset/media/607812
Match each question with the correct measurement, rounded to one decimal place.

What is the approximate area of the triangle?

What is the approximate length of the base of the triangle for the given altitude?

What is the approximate length of the altitude of the triangle?

16.1 units

3.9 units

8.1 units

7.6 units

15.8 units

8.0 units

Found 5 solutions by math_tutor2020, MathTherapy, greenestamps, ikleyn, mananth:
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

Answers:
Area = 15.8 square units
Base = 8.1 units
Altitude = 3.9 units

--------------------------------------------------------------------------

Explanation

Use the distance formula to determine the length of side AB
A = (x1,y1) = (5,2) and B = (x2,y2) = (-3,1)

approximately

Follow similar steps to find the distance from C = (-2,5) to D = (-1.52,1.18) is roughly 3.85004 which rounds to 3.9

Then,
area of triangle = 0.5*base*height
area = 0.5*AB*CD
area = 0.5*8.1*3.9
area = 15.795
area = 15.8 square units.

There's rounding error going on here.
The area of triangle ABC should be 15.5 square units exactly.
This can be determined in the next section below.

--------------------------------------------------------------------------

When given the coordinates of the vertex points, we could use the shoelace formula to fairly quickly find the area of any polygon.

Given list of vertex points
(5,2)
(-3,1)
(-2,5)

Make a copy of the first point (5,2) and place it at the bottom of the list
(5,2)
(-3,1)
(-2,5)
(5,2)
This will help form a loop.

Space the x and y coordinates out. Each (x,y) point gets its own row.

Then draw in the diagonals as indicated below.

Multiply along the red diagonal pairs and add up those products.
5*1+(-3*5)+(-2*2) = 5-15-4 = -14
Do the same for the blue diagonal pairs.
5*5+(-2*1)+(-3*2) = 25-2-6 = 17
Subtract the results.
red - blue = -14 - 17 = -31
We get a negative result. Let's take the absolute value to get |-31| = 31
Lastly, take half of this to get the area = (1/2)*31 = 15.5

The area of the triangle is exactly 15.5 square units.
GeoGebra can be used to verify.

More practice on the shoelace formula is found here
--------------------------------------------------------------------------

What's another way to see how the area is exactly 15.5?

My post is already quite lengthy so this section will be a brief overview rather than getting into the gritty details.
The distance formula will show that
AB = sqrt(65)

The equation of line AB is y = (1/8)x + 11/8
The equation of line CD is y = -8x - 11
Intersect those two equations to determine point D is at the exact location (-99/65, 77/65)
Note how -99/65 = -1.52 and 77/65 = 1.18 approximately.
So that's where your teacher is getting (-1.52, 1.18) for point D.

Use the distance formula to determine that CD = 31/sqrt(65)

Then lastly,
area = 0.5*base*height
area = 0.5*AB*CD
area = 0.5*sqrt(65)*( 31/sqrt(65) )
area = 0.5*31
area = 15.5
which is exact without any rounding done to it.

Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

ΔABC has vertices at A(5,2), B(−3,1), and C(−2,5). Point D is located on the intersection of the altitude and AB�, in such a way that D has coordinates at approximately (−1.52,1.18). https://cds.flipswitch.com/tools/asset/media/607812 Match each question with the correct measurement, rounded to one decimal place. What is the approximate area of the triangle? What is the approximate length of the base of the triangle for the given altitude? What is the approximate length of the altitude of the triangle? 16.1 units 3.9 units 8.1 units 7.6 units 15.8 units 8.0 units The EASIEST and BEST method to determine the AREA of any POLYGON is the SHOELACE METHOD!

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Interesting problem; and interesting responses....

Finding the area of the triangle using the coordinates of A, B, C, and D in the formula for the area of a triangle (one-half base times height) is not a particularly good way. For one thing the coordinates of D are only given approximately; for another thing, you end up doing ugly arithmetic with decimal approximations of the lengths of the base and height.

On the other hand, since the problem asks to find lengths rounded to the nearest tenth, you can get the answer that way.

But, wait -- using either that approximate method or any of at least two exact methods, the correct answer of (exactly) 15.5 is not one of the answer choices. So here the problem is faulty....

Finally, as a note regarding the response from the other tutor, you should never take it as the gospel truth when someone says that a particular method is the best way to work a problem. Enclosing the triangle in a rectangle and finding the area of the given triangle as the area of the rectangle minus the area of the three small triangles is A good way to solve the problem; but there are other equally good ways.

The length of the base AB is found simply using the Pythagorean Theorem -- although again, given the set of answer choices and the instruction to find the length to the nearest tenth, 8.1 is the obvious answer.

And, finally, only one of the answer choices is reasonable for the length of altitude AD.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

        Here I will explain you how to find quickly the area of triangle ABC
        with the vertices at the integer grid points in a coordinate plane.

Draw the minimal rectangle on the grid which concludes the given triangle.
Identify the triangles on the grid inside this rectangle that are outside the given triangle.

It is quite easy to find the area of each such a triangle.

After that, you will find the area of your triangle ABC mentally, by subtraction
the areas of the three other triangles from the area of rectangle.

Notice that doing this way, You will find the PRECISE area of triangle ABC mentally and quickly,
without doing boring calculations.

It is the best strategy and the most valuable piece of knowledge, which you can learn from this problem.

Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website!

Find AB by distance formula
A(5,2) B(-3,1)
AB = sqrt((2-1)^2+(5-(-3))^2) =sqrt(65) Base of triangle
C=(-2,5) , D(-1.52,1.18)
CD = sqrt((-2+1.52)^2+(5-(1.18))^2) =3.85
Area of triangle = 1/2 *8.06*3.85
15.5 unit^2

Question 1206482: Given the two circles defined by the equations x^(2)-6x+y^(2)+8y=12 and x^(2)+y^(2)=4y, find the algebraic equation of the line connecting their centers.
Found 3 solutions by greenestamps, math_tutor2020, ikleyn:
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

If you need to find the centers and radii of the two circles, then you need to complete the squares.

In this problem, you are to find the equation of the line connecting the centers of the circles. That means you don't need to know the radii of the two circles; and that means completing the square is unnecessary work.

As tutor @ikleyn says, you can find the centers of the two circles mentally:

First circle: (x^2-6x+...)+(y^2+8y+...) = ... ---> center (3,-4)
Second circle: (x^2)+(y^2-4y+...) = ... ---> center (0,2)

Then finding the equation of the line containing those two centers is a basic algebraic process.

Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

Answer: y = -2x+2

Explanation

Let's complete the square for the 1st equation.
x^2-6x+y^2+8y = 12
(x^2-6x)+(y^2+8y) = 12
(x^2-6x+9-9)+(y^2+8y+16-16) = 12
(x^2-6x+9)+(y^2+8y+16)-9-16 = 12
(x-3)^2+(y+4)^2-25 = 12
(x-3)^2+(y+4)^2 = 12+25
(x-3)^2+(y+4)^2 = 37
This circle is centered at (h,k) = (3,-4)
The radius is r = sqrt(37)
The circle template is (x-h)^2+(y-k)^2 = r^2
On the 3rd step, I added and subtracted 9 in the first parenthesis grouping. This is to complete the square for the x terms. I took half of the x coefficient and squared it.
The same idea is used to complete the square for the y terms also.

Now complete the square for the other equation.
x^2+y^2 = 4y
x^2+y^2-4y = 0
x^2+( y^2-4y ) = 0
x^2+( y^2-4y + 4-4) = 0
x^2+( y^2-4y + 4) -4 = 0
x^2+(y-2)^2 -4 = 0
x^2+(y-2)^2 = 4
(x-0)^2+(y-2)^2 = 4
The center here is (0,2). The radius is 2.
On the 4th step, I took half of the y coefficient and squared it. That explains the +4-4 part at the end of the parenthesis grouping.

The two centers we found were
A = (3,-4)
B = (0,2)

Let's find the equation of line AB.
First we need the slope.

The slope of line AB is m = -2.

The y intercept is b = 2 due to the point (0,2).

The equation of line AB is therefore y = -2x+2 since the slope-intercept template is y = mx+b.

You can use a tool like GeoGebra to confirm the answer.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Given the two circles defined by the equations x^(2)-6x+y^(2)+8y=12 and x^(2)+y^(2)=4y,
find the algebraic equation of the line connecting their centers.
~~~~~~~~~~~~~~~~~~~

To find the centers, apply completing the squares separately to x-terms and y-terms in each equation. It can be done MENTALLY. The center of the 1st circle is the point (3,-4). The center of the 2nd circle is the point (0,2). The slope of the line through the centers is m = = = -2. So, an equation of the line can be presented in the form y-2 = m*(x-0), or y - 2 = -2x, or y = -2x + 2, or in any other equivalent form. You can check on your own that the presented equations are satisfied with the coordinates of the centers, so this straight line goes through these points.

Solved.

Question 1206483: A bug lives on a corner of a cube and is allowed to travel only on the edges of the cube. In how many ways can the bug visit each of the other seven corners once and only once, returning to its home corner only at the end of the trip?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

See the solution 2 under this link

https://artofproblemsolving.com/wiki/index.php/2006_AMC_12A_Problems/Problem_20

They say that if you are able to understand it, then you are officially good in Math.

Question 1074065: The segment AB is one third of the entire length of segment AD. If A(3,-2) and B(0,2), then where is the terminal point D located?
Found 3 solutions by ikleyn, greenestamps, mananth:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The segment AB is one third of the entire length of segment AD.
If A(3,-2) and B(0,2), then where is the terminal point D located?
~~~~~~~~~~~~~~~~~~~~~

By the way, segment AD may have an OPPOSIITE direction comparing with AB,

or any other direction on the plane, different from that of AB, so this problem,
as it is worded in the post, is HEAVILY DEFECTIVE.

From the written post, it remains unclear how the point B does relate to the segment AD.

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Of course, we don't HAVE to use the formula shown by the other tutor. If you want to solve the problem using formal mathematics, there are alternative forms of the formula that make it easier to understand.

But solving the problem informally using common sense is MUCH easier.

B is one-third of the distance from A to D.

In the x direction, from A to B is a change of -3, so from A to D the change in the x direction is 3(-3) = -9. So the x coordinate of D is 3+(-9) = -6.

In the y direction, from A to B is a change of +4, so from A to D the change in the y direction is 3(+4) = +12. So the y coordinate of D is (-2)+12 = 10.

ANSWER: D(-6,10)

Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website!
We have to use section formula for internal division
if we have a point P(x,y) that divides the line segment with marked points as A (x1,y1) and B(x2,y2). To find the coordinates, we use the section formula, which is mathematically expressed as:
P(x, y) = (mx2+nx1)/(m+n), (my2+ny1)/(m+n)
.
A= (x2,y2)~(3,-2)
B= (x,y) ~(0,2)
m=1 , n=2
0= (1(x)+(2)(3))/3
x=-6
2= ((1)(y)+(2)(-2))/3
6= y-4
y=10
x(-6,10)

Question 1115219: T(-10,-2) R(10,8) Y(11,-9) are vertices of ∆TRY
1) find equation of YA, the altitude from Y to TR . show that A(4,5)
2) if equation of perpendicular bisector is y=3x-7, find equation of circle through T, R, Y .
3) find the area of ∆TRY
4) find size of angle T
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website!
Equation of TR
x1 y1 x2 y2
-10 -2 10 8

slope m = (y2-y1)/(x2-x1)
( 8 - -2 )/( 10 - -10
( 10 / 20 )
m= 0.50

Plug value of the slope and point ( -10 , -2 ) in
Y = m x + b
-2.00 = -5 + b
b= -2.00 - -5
b= 3
So the equation of TR will be
Y = 1/2 x + 3 ( m=1/2)..................................(1)
AY is perpendicular to TR y=(11,-9)
slope of AY = -2 ( negative reciprocal)
Equation AY
slope =2 , Y=(11,-9)
(y-(-9))= (-2)(x-11)
y+9= -2x+22
y= -2x+13.................................................(2)
Solve (1)&(2) to get point A (4,5)
.
--------------------

Question 1206377: A machinist in a fabrication shop needs to bend a metal rod at an angle of 60° at a point 4m from one end of the rod so that the ends of the rod are 12m apart.
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
This suggests to use Law Of Cosines to find some value t, the distance from the bend point
to the other end point of the rod.
Draw the figure described in your problem description.

One end point given as 4 meters from one end point and t is distance from bend point to the
other end point of the rod.

Solve for t.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A machinist in a fabrication shop needs to bend a metal rod at an angle of 60°
at a point 4m from one end of the rod so that the ends of the rod are 12m apart.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Do you have a question ?

Question 1205759: Determine if the angle is complementary, supplementary, or vertical. Find the value of x in the figure.
angle AB = (x+8)
angle BC = 2x
What is the value of x?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

The diagram is missing. Also it's a bit strange how your teacher named each angle using two letters. Normally 3 letters are involved.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

The figure is missed/lost in this post, making the solution IMPOSSIBLE.

Question 1205320: Question Number 11 of 20 - Geometry
Circle O with radius 10.
The area of the shaded sector of circle O
is -
f 50π
g 25π
h 5π
j 20π

Answer by MathLover1(20855)   (Show Source):
You can put this solution on YOUR website!

what is the shaded sector of circle O
the diagram of circle O is missing, you need to post it