Questions on Geometry: Triangles answered by real tutors!

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if you have an isosceles triangle with the legs being 9 and the base is 10 find the altitude.
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Draw the picture:
Draw a perpendicular bisector to the base: (that is the height)
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You have two right triangles with base = 5 ; height = "h" ;
and hypotenuse = 9
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Pythagoras:
9^2 = 5^2 + h^2
h^2 = 81-25 = 56
h = sqrt(56) (that is the altitude of the isosceles triangle)
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Cheers,
Stan H.

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height of isoscles triangle = sqrt(a^2-(b/2)^2)

where a is the length of the leg
and b the base





h=8.64

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Many different ways. The most common one is
(b = base, h = height)

For the next several formulas, denote

a,b,c = side lengths
A,B,C = angles opposite from sides a,b,c
s = semiperimeter (=(a+b+c)/2)
r = inradius
R = circumradius

Other area formulas you'll see are



(Heron's formula)

where AB and AC are the vectors from A to B and A to C respectively





(excuse my sloppy notation of using A twice...treat them as different variables)

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The area of an equilateral triangle = where a is the length of the side
length = 12

Area =

Area = 62.3 cm^2

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use the longest side as the base

the altitude (h) divides the triangle into two right triangles; a, h, 30 and b, h ,23

a + b = 33

a^2 + h^2 = 30^2

b^2 + h^2 = 23^3

three equations, three unknowns

subtracting 2nd and 3rd equations ___ a^2 - b^2 = 30^2 - 23^2 ___ (a + b)(a - b) = 371

dividing by 1st equation ___ a - b = 371 / 33

adding 1st equation ___ a - b + a + b = 33 + (371 / 33) ___ 2a = 1460 / 33

solve for a, then substitute back to find h

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the "Law of Sines" states that the lengths of the sides of a triangle are in the same ratio as the sines of the angles opposite the sides

side x is opposite angle X

side y is opposite angle Y

x / y = sin(X) / sin(Y)

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find ar if ab is 18 milimeter
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Not enough info

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If the ratio of lengths of corresponding sides is the same, the triangles are similar. So we match sides, shortest with shortest, longest with longest, and medium with medium.



The triangles are similar.

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The sides of a triangle are 30, 40 and 50. What is the ratio of the sides of the triangle?
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they're 30:40:50
or 3:4:5

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if you have an isosceles triangle with both legs equaling 27 and both base angles measuring 23 degrees will the base side also equal 27?
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No.
If the 3 sides are equal, then all 3 angles = 60 degs.

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half the base equals 8
a^2+b^2=c^2
8^2+b^2=16^2
64+b^2=256
b^2=256-64
b^2=192
b=sqrt 192
b=13.9 or 14

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find the area of a 30-60-90 triangles with longest leg of 12
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The longest leg must be opposite the 60 degree angle.
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Let the shortest leg be "x".
Then the hypotenuse is "2x".
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Equation:
x^2 + 12^2 = (2x)^2
x^2 + 144 = 4x^2
3x^2 = 144
x^2 = 144/3
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x = 12sqrt(3)/3
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x = 4sqrt(3)
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Area = (1/2) leg*leg
Area = (1/2) 12*4sqrt(3)
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Area = 24sqrt(3) square units
=================================
cheers,
Stan H.
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a) 4,5,10 because 4+5 is less than 10, but the triangle inequality states that it has to be greater than 10.

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Solve the equation for which variable?

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AB and ED are the poles (perfectly vertical). BE and DA are the ropes that cross at C.
F is the point directly below C on the ground (line AE), which is pefrectly flat and horizontal.
The vertical poles are part of parallel lines.
As a consequence, triangles ABC and DEC have congruent angles at B and E, and at A and D (alternate interior). Of course, ABC and DEC also have congruent angles at C (vertical angles).
Triangles ABC and DEC are similar, with corresponding sides in the ratio 2:1

In particular,
and
Right triangles ABE and FCE, with the same angle at E, are also similar, so
--> -->
The ropes cross 4 ft above the ground.

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hypotenuse = 17
for tan you need to know the opposite & adjacent sidex
Cos 54 = adjacent side / hypotenuse
cos 54 * hypotenuse = adjacent side
adjacent side = 9.99
opposite side/adjacent side = Tan
9.99 * tan(54 degrees) = 13.7500554

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The perimeter of a triangle is 23 feet. Two sides are equal and the length of the third side is 1 ft less than twice the length of an equal side. What is the length of one of the equal sides?
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Equation:
x + x + (2x-1) = 23
4x - 1 = 23
4x = 24
x = 6 ft (length of 2 of the sides)
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2x-1 = 2*6-1 = 11 ft (length of the 3rd side)
==================================================
cheers,
Stan H.

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how would i be able to figure out if something is a triangle for example if they would give me {8,24,25}?
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Figure out what?
Area?
Perimeter?
Angles?

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triangle FGK is similar to triangle BGC by SAS.
FG is proportional to BG in the same ratio that KG is proportional to CG.
angle KGF is congruent to angle BGC because they are the same angle.

since these triangles are similar, their corresponding angles are congruent.
congruent angles have equal measure.

you are given that angle ABG is equal to 130 degrees.
this means that angle GBC is equal to 50 degrees because these angles are supplementary.

angle GFK is the corresponding angle in triangle FGK, so angle GFK is also equal to 50 degrees because the triangles are similar.

since angle GFK is equal to 50 degrees, this means that angle DFC is also equal to 50 degrees because angle GFK and angle DFC are vertical angles and vertical angles are equal.

since angle DFC is equal to 50 degrees and angle FDC is given as being equal to 20 degrees, this means that angle FCD is equal to 110 because the sum of the angles of a triange is equal to 180 degrees.

since angle FCD is equal to 110 degrees, this means that angle GCB is equal to 70 degrees because angle FCD and angle GCB are supplementary.

since angle GCB is equal to 70 degrees, this means that angle GKF is equal to 70 degrees because angle GKF is the corresponding angle in triangle FGK to angle GCB in triangle BGC.

since angle GKF is equal to 70 degrees and angle GFK is equal to 50 degrees, this means that angle KGF is equal to 60 degrees because the sum of the angles of a triangle have to be equal to 180 degrees.

since angle KGF is equal to 60 degrees, this means that angle BGC is also equal to 60 degrees because it's the same angle.

angle GKJ is supplementary to angle GKF which means that angle GKJ is equal to 110 degrees since angle GKJ and angle GKF are supplementary which means that the sum of their angles has to be equal to 180 degrees.

note also that angle GKJ is an exterior angle to triangle GKF which means that it is equal to the sum of the remote interior angles of triangle GKF. these angles are KGF and GFK whose sum is 60 + 50 = 110 degrees.

the following diagram should support the logic just given.


in this diagram, triangle FGK was turned around so you can see the corresponding angles and corresponding sides easier.

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You know that triangles ABC, DCL, CEG, etc. are all similar due to AA similarity. Furthermore, the lengths of the sides of the triangles are in the ratio 3:4:5, with 5 being the hypotenuse. Use this ratio to find the side lengths.

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John

My calculator said it, I believe it, that settles it

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A = (bh)/2

54 = (12h)/2

54*2 = 12h

108 = 12h

108/12 = h

9 = h

h = 9


So the height is 9
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Here is the way to analyze the situation. Since Z is opposite of the obtuse (biggest) angle, Z must be the longest side of the triangle. (This should be obvious from the drawing you were given for the triangle.)
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That means that the SUM of lengths of the two sides labeled X and Y must be greater than Z. (I believe that this is a theorem or an axiom.) Why is that true? Think about this. You cannot form a triangle if the lengths X and Y when added together are less than or just equal to the length of Z. If you need help picturing this, make this drawing:
.
A. Draw a 4 inch long line about in the center of a sheet of paper.
B. Take a drawing compass and set the distance between its point and lead tip at 2 inches.
C. Place the compass point at one end of the 4 inch line swing a circle (of 2 inch radius) around that end of the line.
D. Then reset drawing compass so that the distance between its point and lead tip is 1 inch.
E. This time set the compass point at the other end of the 4 inch line and swing a circle (of 1 inch radius) around this end of the 4 inch line.
Notice that this is the case where the lengths of the two short sides (2 inch and 1 inch) total less than the long side (4 inches). The circles do not intersect meaning that the proposed length of 2 inches and 1 inch cannot be joined to form a triangle. If the radii of the two circles added up to be bigger than the longest side (for example one circle of radius 3 and the other of radius 2) then the circles would intersect and a triangle could be formed by drawing lines from the ends of the 4 inch line to the point where the circles intersected. One line to the intersection would be 3 units long and the other line would be 2 inches long, making a triangle of sides 2, 3, and 4.
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Once you can visualize and are convinced that X + Y has to be greater than Z for a triangle to be formed, you can write the inequality for a triangle having Z as its long side:
.

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Then you are told in the problem that:
.

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Solve this given equation for Y in terms of X by multiplying both sides by X to get:
.

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Now go to the inequality that we wrote above and in place of Y substitute its equal 3X to make the inequality become:
.

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Combine the two terms on the left side and you have the requirement that in the given triangle:
.

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This is answer a. in the list of answers that you were given.
.
I hope this helps you to understand the problem that you were given and gives you a little more insight into how triangles are formed.
.

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A triangle has two sides of lengths 4 centimeters and 6 centimeters. Its area is
N square centimeters , where N is a prime number. What is the greatest possible
value of N?
Choose one answer:
a. 12
b. 11
c. 9
d. 2

You can rule out choices a. 12 and c. 9 immediately because they
are not prime numbers.  You can also rule out choice d. 2, because
when two sides of a triangle are given, the third side must be greater 
than the difference of the other two sides, and 2 is equal to 6-4, and it
would have to be greater than 2.  So that only leaves b. 11, which is a
prime number.  

However, it may not be satisfactory to do it entirely by eliminating
the wrong choices.  Here is how you would find the answer if you 
had not been given any choices at all:



Area = ×base×height

N = ×6×h

N = 3h

 = sin(q)

h = 4sin(q)

And since N = 3h

N = 3×4sin(q)

N = 12sin(q)

Divide both sides by 12

 = sin(q)

The sine of an angle must be 1 or less.  Therefore the largest
prime number N could be and  be less than or equal 1 
is when N = the prime number 11.

Answer: b. 11 

Edwin

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When you have two sides of a triangle given, the third side must be

1. Greater than the difference of the two given sides.
and
2. Less than the sum of the two given sides.

So when you have given sides of 9 and 16, the third side k must be:

1. Greater than the difference 16-9 or 7.

and

2. Less than the sum 16+9 or 25.

So that means that 

       7 < k < 25

On the choices, I. 6, II. 16, and III. 25

only II. 16  lies between 7 and 25

You cannot draw a triangle with sides 9, 16 and 6 because the two sides 9 and 6
can only span lengths less than their sum 16.

You also cannot make a triangle with sides 9,16 and 25 because the sides 9 
and 16 can only span lengths less than, not equal to, ther sum 25.

However you can draw a triangle with sides 9, 16, and 16.  It will, of
course be isosceles.

Edwin

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height of an equilateral triangle =
where a is the side. all sides are congruent

H= cm

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A 16 foot ladeer leans against a wall at a 74degree angle of elevation find the distance from the ladder's base to the wall.
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= 16*cos(74)

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What is the angle between the horizontal (x) axis and the line containing the points (2,8) and (4,9)?

We observe that the angle q that the green line makes with the 
x-axis is the same size angle as the angle marked q in the little right
triangle up where the two given points are.

In that little right triangle we see that the opposite side of q is 
1 unit long and the adjacent side to q is 2 units long, 
so we use the trig function that involves opposite and adjacent:

                 tan(q) = 
                 tan(q) =   

Use the inverse tangent fuction on the calculator and get

                 q = 26.56505118° = 26°33'54"

I hope you observed that the tangent of that angle q is the slope
of the green line.
                                                      
Edwin

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We have , that is, for every 100 ft in horizontal distance, the altitude changes by about 5.24 ft, so the grade is 5.24%. Here, it is appropriate to round to 5%.

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one side of an equilateral triangle increased by 11 is 33 less than three times another side. if the sides are measured in inches, what are the measures of the three sides of the triangle?
:
x = the length of the 3 sides of the equilateral triangle
:
"one side of an equilateral triangle increased by 11 is 33 less than three times another side."
x + 11 = 3x - 33
11 + 33 = 3x - x
44 = 2x
x = 22 inches

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Hilly areas often have road signs giving the percentage grade for the road. A 5% grade, for example, means that the altitude changes by 5 feet for each 100 feet of horizontal distance.
Suppose an uphill road sign indicates a road grade of 3%. What is the angle of elevation of the road?
A 3% grade means the altitude changes by 3 feet for each 100 feet.
Applying trigonometry,
tan x = 3/100
tan x = .03
x = 1.72 degrees

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What angle does the line y=7/6x in the xy plane make with the positive x axis?
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If you mean y = (7/6)x, then
m = 7/6 = arctan(angle)
angle =~ 49.4º

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I agree with the other tutor, read your post again. If that second "is" should be "it" then it doesn't take much to determine whether it is a triangle.