Questions on Geometry: Triangles answered by real tutors!

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with the givens we know that PQ = PR, because sides opposite congruent angles are equal.
now square both sides and we will eliminate the squareroot


now factor

so x=1 and -2...throw out the negative value
and x=1 and PR=2

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with the givens we know that PQ = PR, because sides opposite congruent angles are equal.
now square both sides and we will eliminate the squareroot


now factor

so x=1 and -2...throw out the negative value
and x=1 and PR=2

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First, to draw an isosceles right triangle, draw a square, then draw one of its diagonals. You now have two congruent isosceles right triangles.
Now draw the other diagonal and you will see that the diagonals of a square bisect each other.
Next, erase one of the triangles (that's half the square) and you are left with one isosceles right triangle whose height (h) is exactly half the length of its base (b). But the base is really the hypotenuse of this right triangle. So, for the area of this triangle, you can write:
But , so substitute this to get:
Simplify.
where b is the length of the hypotenuse.

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I have a triangle where the angles are given as 46 degrees, x and y. The sides adjacent to y are equal to each other. I am asked to solve for angles x and y
:
I can't believe how I screwed this one up, it should be:
:
If the sides adjacent to y are equal, it's an isoceles triangle; therefore
angles 46 and x are equal:
x = 46 degrees also
y = 180 - 46 - 46
y = 88 degrees
:
I am sorry! Carl

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why must the sum of two sides of a triangle always be greater than the third side.

Because the shortest distance between two points 
is the line segment joining them.




The shortest distance from point A to point B 
is the segment AB. So it certainly farther to go 
from A to B by going to C first and then from C to B.

Edwin

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if the area of a triangle is 60 sq m and the base is 20 m, what is the altitude h.
.
For any triangle,
area = (1/2)bh
where
b is length of base
h is height
.
Plug in what was given into:
area = (1/2)bh
60 = (1/2)(20)h
solve for h:
60 = (1/2)(20)h
60 = (10)h
6 m = h (altitude)

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Find the values of the variables:
one angle = 90 degrees
one angle = (2x-11)
one angle = (y+16)
.
Since you have two unknowns (x and y), you'll need two equations.
.
equation 1: (as you say)
(2x-11)+(y+16) = 90
.
since there are two congruent sides:
equation 2:
(2x-11)=(y+16)
.
Solve equation 2 for x:
(2x-11)=(y+16)
2x-11 = y+16
2x = y+27
x = (1/2)(y+27)
.
Plug the above into equation 1 and solve for y:
(2x-11)+(y+16) = 90
(2(1/2)(y+27)-11)+(y+16) = 90
y+27-11+y+16 = 90
2y+27-11+16 = 90
2y+16+16 = 90
2y+32 = 90
2y = 58
y = 29
.
Plug it back into equation 2 to find x:
(2x-11)=(y+16)
(2x-11)=(29+16)
2x-11=45
2x=56
x = 28
.
solution:
x is 28
y is 29

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JM=MN
JM=2X-5
MN=3x-9
JN=x-2
Thus: 2x-5=3x-9
2x-3x=-9+5
-x=-4
x=4 answer.
2*4-5=8-5=3 for JM.
3*4-9=12-9=3 for MN.
4-2=2 for JN.

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Its hard to know what you are doing without the full instructions and what "20x+10, 30x -2, 7x+1 " are referring to (are these side lengths? angles?)

However, since I'm seeing a "360" I'm assuming that you are working with angles.

There's a problem though: ALL angles in a triangle add to 180 degrees (not 360 degrees). So the equation should be:





Let me know if you still need help.


Note: the answer is and the three angles are 70, 88, and 22

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On second thought:
If the natural number is a multiple of three, it can be written
as a sum of three numbers, each of which is one-third the original
number. If that sum is not used, one of the numbers in the sum
must be less than one-third the original number.
------------
If the natural number is not a multiple of three, at least one
of the numbers in the sum must be less than one-third the original
number.
------------
Stan H.
Cheers,

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---> let's point it as the adjacent side
-------------> let's point it as the oppposite side
-------------> hypotenuse side
differ by 2cm --
By Pyth. Theorem:



, perfect square, factorable

2 values ---
As highlighted, shortest side: adj= 6cm; opp=6+2=8cm; hyp=6+4=10cm
See below,

check:



Thank you,
Jojo

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I'll leave the sketch with you okay, & hopefully we can follow together:
We find first the
where, , &
So,

Now , next is interesting in finding CE?
We first get by trigo function of right triangle. How?
Since AD split the base(BC) in to half, right? Same as .
That forms a right triangle (ADB).



.
Now we know line CE(?) cuts line AB into half:, right?
That forms a triangle and we'll use Cosine Law in getting line CE.

where ---
Continuing,




Thank you,
Jojo

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You have done ok so far!
In your triangle, you have , , and .
To see how we get b, you have:
Take the square root of both sides.

The sin of theta (I'll call it A because I can't do a theta symbol) is given as because the sine of the angle is the side opposite the angle (a = 1) over the hypotenuse (c = 2), so you get
The cosine of this angle (A) is the side adjacent to the angle ( over the hypotenuse (c = 2), so...

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Given:Triangle ABC
Segment BD is perpendicular to segment AC
Segment BD is a median of Triangle ABC
What kind of triangle is Triange ABC?
BD is perpendicular to AC. This means that angle D is 90 degrees.
Two triangles are formed: Triangle ABD and Triangle CBD.
BD is also a median. This means that it divides AC into two equal parts.
From the given data, I gather that AB = BC but also that AB and BC do not equal AC.
I think this is an isosceles triangle.

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Here's a quick note about what the ratio is really saying:

Since the ratio is vertex angle A to the measure of angle B, this means that we have this ratio:

A:B ---> 2:5

So this means that the vertex angle is SMALLER than the base angle. However, if you have a base angle of 40 degrees, this means that the vertex angle is which means that the vertex angle is larger (not smaller) than the base angle. So this effectively rules out the "2nd Approach" answer. Remember, the ratio is vertex:base angle (not the other way around)


However, I agree with your first approach. That answer is correct. You'll find that the base angles are 75 degrees and the vertex angle is 30 degrees. Since the ratio of 30:75 reduces to 2:5, this verifies the answer.


Note: it is IMPOSSIBLE to have both an isosceles triangle AND have a base angle that is greater than 90 degrees. Why? Base angles are equal. So if there is an obtuse base angle, there are really 2 obtuse angles in the triangle (which is impossible; triangles can only have at most one obtuse angles). So this means that either the book is wrong or maybe you read the wrong answer.

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32 is correct.

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A rectangular piece of cardboard with perimeter 30 in., two parallel and equally spaced creases are made. The cardboard is then folded to make a prism with open ends that are equilateral triangles. Find the volume of the prism as a function of x and what is the domain of V(x)?
:
Let x = the 3 equal distances, between the two folds
then
3x = width of the cardboard
and
Let L = the length of cardboard
;
Perimeter:
2(3x) + 2L = 30
Simplify divide by 2
3x + L = 15
L = (15-3x)
:
Find the height (h) of the equilateral triangle with sides = x
h^2 + (x/2)^2 = x^2
h^2 = x^2 - (x/2)^2
h =
:
Vol of a triangular prism = *L*w*h
In this problems
L = (15-3x)
w = x
h =
:
V(x) = *(15-3x)*x*
:
You can tell by L, (15-3x), that x has to be less than 5 (the domain)
:
If you graph this:

The max volume when x = 3.33, Domain 0 to <+5

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sqrt(a+5)^2=a+5 answer.

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There are 2 possibilities.
If c is the hypotenuse, the 3rd side is 5 (sqrt(c^2 - b^2)).
The 3rd side could be sqrt(c^2 + b^2), or sqrt(33), too.

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isosceles means that the two base angles are equal as are the sides opposite the angles

LW=TW __ 7X+21=4X+21 __ subtracting 4x+21 __ 3X=0 __ X=0

LW=TW=21

there are 180º in a triangle __ the vertex angle is 90º, so the sum of the two base angles is also 90º
__ so each bse angle equals 45º

by Pythagoras __ LT^2=LW^2+TW^2 __ LT^2=2(21^2) __ LT=21*sqrt(2)

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As the hypotenuse is the longest side of a right triangle, so
the hypotenuse c = 5.
The length of one of its leg is a = 3
So the length of the other leg is
b =
Thus the area =

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I'll do the equation.
Solve for h.
A = 1/2 (bh)
We can write 1/2 (bh) like bh/2 for easy reading. They both mean the same thing.
We now have A = bh/2
Multiplying both sides of the equation by 2 and we get:
2A = bh
To solve the equation for h, we want to get h alone.
To do so, divide both sides of the equation by b.
2A/b = h

Did you follow?

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Do you have a sample question related to coordinates?
Post your question in full and then we'll be able to guide you safely through.

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Let's call the two angles A and B.
1.
You also know that


Use eq. 1 and substitute into this equation,





The acute angle with the greater measure is 60 degrees.

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I agree with your answers.

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x + x + 1 + x + 2 = 12
3x +3 = 12
3x = 9
x = 3
x+1 = 4
x+2 = 5

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if i understand this correctly, it is
-3 - 4(t-5) = -2(t+3) + 11
there are different ways to approach it, but this way works on this equation.
-----
first multiply out to remove the parentheses.
-3 -4t + 20 = -2t - 6 + 11
combine like terms.
-4t + 17 = -2t + 5
add 4t to both sides of the equation.
17 = 2t + 5
subtract 5 from both sides of the equation.
12 = 2t
divide both sides of the equation by 2.
t = 6
-----
to prove, substitute in original equation
-----
-3 - 4(6-5) = -2(6+3) + 11
simplify expressions within parentheses.
-3 -4(1) = -2(9) + 11
perform operations to remove parentheses.
-3 -4 = -18 + 11
combine like terms
-7 = -7
-----
answer is proven correct.
answer is t = 6
-----

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picture johnstown as the center of a circle

smithville is on a circle with a 70 mi radius and tomstown is on a circle with a 90 mi radius

if they are on the same side of the circles (in the same direction from johnstown)
__ the closest they can be is 20 mi (90-70)

if they are on opposite sides of the circles (opposite directions from johnstown)
__ the farthest apart they can be is 160 mi (90+70)

there is only one answer between 20 and 160

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It's hard to answer without an example to work with. Why don't you post a problem, and maybe we can help you with that.
:)

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Okay, here we go. Not a difficult problem, but it comes in five parts. Here we go…
First of all, you must draw your picture correctly according to the description:
First, draw an isosceles triangle, but don’t worry about the dimensions, as long as you determine which side is the base. Now label it correctly. You have the base on the bottom, labeled F and G.
The top of the triangle is E. Now you are given the two midpoints of the sides. Now, a midpoint divides the side in question into two equal parts, and since this is an isosceles triangle, both of the sides of the triangle (EF and EG) are equal, and thus, since each of two equal sides are cut exactly in half, each of those 4 segments are equal. Now we can answer the first part:
WHAT IS THE VALUE OF x?
Since each of the four segments are equal, and the lengths of two of them are given (2x+3 and 5x-9), we can set these equal to each other to solve for x:
So, 2x+3 = 5x-9. Solving for x gives you that x=4. That is the answer to the first question.
FIND THE PERIMETER OF TRIANGLE EFG.
Since we now know that x=4, then the lengths of each of the side segments is 11, and thus each of the equal sides of the isosceles triangle is 22. You’re also given that the base is equal to 4x, and since we know that x=4, we know that the base is 16. Since the perimeter of a triangle (or any figure) is the sum of the sides, then the perimeter of this triangle is 22 + 22 + 16, or 60.
FIND THE AREA OF THE TRIANGLE USING PYTHAGORAS. ROUND TO THE NEAREST WHOLE NUMBER.
The Pythagorean theorem says that, in a right triangle, the square of the hypoteneuse is equal to the sum of the squares of the other two sides.
Thus, we have to find the height. The math to prove this is too long to show here, but suffice to say that the line dropped from the vertex of the third angle of an isosceles triangle does a few things. It bisects that angle, it forms a right angle with the base, and it bisects the base. Since it bisects the base, and we know that the base = 16, each side of the base on either side of the right angle = 8. We then now know two of the three sides of the right triangle. The hypoteneuse is 22, the base is 8, and, thus, we can compute the value of the third side, which is the height of the big triangle.
So, we have that 8^2 + x^2 = 22^2.
This gives us 64 + x^2 = 484. Subtract 64 from both sides, and this gives us:
x^2 = 420. Now take the square root of both sides and this gives us that:
x = 20.49.
Now we have the base and the height forming a right angle with the base. We can then use the formula to find the AREA of the triangle using the formula:
AREA = ½ (BASE)(HEIGHT):
AREA = ½ (16)(20.49) = 163.95 = 164 (the answer was asked to be rounded to the whole number).
FIND THE AREA OF THE TRIANGLE USING HERO’S FORMULA
Hero’s formula says that if S = (the sum of the sides of the triangle) divided by 2, then the AREA of the triangle can be given by:
AREA = the square root of [S (S – side one)(S – side 2)(S – side 3)]
So, in this case, S = (22 + 22 + 16)/2, which is 60/2, or 30.
Thus the AREA = the square root of [30(30 – 22)(30 – 22)(30 – 16)], which is sqrt[30(8)(8)(14)], which is sqrt[26880], which is 163.95 = 164 (the answer was asked to be rounded to the whole number).
HOW DO THEY COMPARE?
They are the same.
WHEW !!! Hope that this helps. Good luck !

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3s+3s+s=21
7s=21
s=3 For the third short side.
3*3=9 for the equal sides.
Proof:
9+9+3=21
21=21

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if you know the area of a triangle = 60 sq feet and the base is 6ft, what's the height?
.
Because the area of ANY triangle is:
A=(1/2)BH
where
A is the area
B is length of the base
H is the height
.
Plugging in what we know:
60 = (1/2)(6)H
Solving for H, start by multiplying both sides by 2:
120 = 6H
Dividing both sides by 6:
20 feet = H (height of triangle)

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You use the Pythagorean theorem to solve.

a=5
c= 12



take the square root of both sides;
bc = 10.91
:)

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See this --> http://www.mathopenref.com/triangle306090.html
If the longer leg is 6, that corresponds to the side
The short leg is
The hypotenuse is

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If the base of a larger triangle is 34 inches long, what is the length of side A of the smaller triangle? The smaller triangle's base is 17.. im so not good at math and have been stuck on this for three days someone please help me asap.. this is due later today..
-------------------
That's not enough info. All that can be determined from that is that the side A of the smaller is 1/2 that of the larger, but it's necessary to assume they're similar triangles.
Is there additional info, or a drawing?

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i need help
--------------
I don't have that book, so I can't help.

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a+b+c=180

"angle a is 4 times greater than angle b" __ a=4b

"angle c is 20 degrees less than angle b" __ c=b-20

substituting __ (4b)+b+(b-20)=180 __ 6b=200 __ b=100/3

substituting __ a=4(100/3) __ a=400/3

substituting __ c=(100/3)-20 __ c=40/3

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A complete circle is 360 degrees
A half-circle is then 180 degrees

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height = h
base = h+7
area = 60
-----
formula for area = 1/2 base * height
base = h+7
height = h
area = 60
-----
formula becomes
(1/2)*h*(h+7) = 60
multiply both sides by 2 to remove the denominator.
h*(h+7) = 120
multiply out to remove parentheses
h^2 + 7*h = 120
subtract 120 from both sides of equation
h^2 + 7*h - 120 = 0
this is a quadratic equation with factors equal to
(h+15) * (h-8) = 0
-----
h = -15 no good because can't be negative.
answer is
h = 8
h+7 = 15
-----
1/2 base * height should = 60.
1/2 * (15) * (8) = 4 * 15 = 60.
answer is good.
h = 8.

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triangle xyz
xy = yz
angle x = 52 degrees.
draw triangle with x on the left, y on top, z on the right.
xy is line going up from left to right.
yz is line going down from left to right.
angle x is the angle in the left hand corner of the triangle.
this is an isosceles triangle because 2 sides are equal (congruent).
in an isosceles triangle the base angles are equal.
these are the angles opposite the equal sides.
angle x is opposite yz
angle z is opposite xy
angle z is therefore equal to angle x is equal to 52 degrees.
since the sum of the interior angles of a triangle is always 180, then angle y must be equal to 180 - angle x - angle z.
this becomes angle y = 180 - 2*52 = 180 - 104 = 76.
angle y must be 76 degrees.
to prove add all 3 angles of the triangle and they must equal 180 degrees.
52 + 52 + 76 = 180.
-----
angle y is 76 degrees.