If a triangle has a side lenghts of 6, 8, and 10, find the measurement of each angle

Question

Need to know all angles

Answer 1

6, 8, and 10 are a Pythagorean triple, so this is a right triangle (6² + 8² = 10²) and one of the angles is 90^o.  The other two angles are:

 

tan^-1(4/3) = 36.9^o

tan^-1(3/4) = 53.1^o

 

36.9^o+53.1^o=90^o, as expected.

Answer 2

If you look at the sides you can see that they are in a ratio of 3,4,5.

This is a 3,4,5 triangle

- one of the special triangles that never gets put in the book

with the 45-45-90 and 30-60-90.

So you can see this is a right triangle with the long (10) side as the hypotenuse.

Draw a picture of it and it will help you.

One angle, call it C, is 90º.

 

Since you have all 3 sides you can find the other 2 angles with any of the trig functions you want.

How about tan^-1(3/4)= A

How about tan-1(4/3)= B

 

I'll let you take it from here.

Answer 3

this is basically a 3-4-5 right triangle

the legs are 6(2*3) and 8(2*4), and the hypotenuse is 10(2*5)

6²+8²=10²

36+64=100

100=100

draw a right triangle with the 90º angle on the left, horizontal leg=8 and vertical leg =6 (you can draw it any way you like; I'm just giving you an example to go by)

the 2 acute angles are at the top left and bottom right

you need to find these angles

using the sine function...

the sine of the bottom right angle is sin=opposite/hypotenuse

sin(x)=6/10

sin(x)=0.6

now using your calculator or a table of values, find the arcsin of 0.6; in other words the sine of what angle is 0.6 ?

arcsin(x)=0.6

x=36.86989765º

you can subtract this angle from 90º to get the other acute angle or use the arcsin function again

this time you are using the arcsin function for the top left angle

sin=opposite/hypotenuse

sin(x)=8/10

sin(x)=0.8

arcsin(x)=0.8; the sine of what angle is 0.8 ?

using your calculator or a table or the web...

arcsin(x)=0.8

x=53.13010235º

you'll notice that adding these two angles with all the decimal places add up to 90º

now you have the measures of the three angles

if you like you can express the angles in degrees, minutes, and seconds

Answer 4

This triangle is part of the Pythagorean triples because it corresponds to a 3,4,5 triangle.  If I multiply these sides by 2 correspondingly, I will get 6, 8, 10.  This means that one angle of this triangle is 90 degrees.

 

Next is to use law is sine to find the second angle.  This will work because we have all three sides and one angle. Pick length 10 in this law of sine because the largest side is the hypotenuse and the hypotenuse is opposite to the right angle of a right triangle.

 

(10 / sin 90) = (8 / sin x)

 

Cross multiply.

 

8 sin 90 = 10 sin x

8 (1) = 10 sin x

8  = 10 sin x

 

8/10 = sin x

 

sin^-1(8/10) = x

 

53.13 = x

 

The angle opposite to length 8 is 53.13 degrees.

 

 

To find the 3rd angle, we use complementary angles.  Subtract 53.13 from 90.

 

90 - 53.13 = 36.87.

 

 

Our angles are

 

36.87° , 53.13° , and 90°.

Answer 5

First notice that 6² + 8² = 10² so what type of triangle is this?

If you know all sides, and one angle, then you can use the law of sines to find the other angles.