SOLUTION: The sides of an isosceles triangle have lengths of 10, 10, and 4. Find the measures of the base angles. Round your answers to the nearest tenth. Show the equation used.

Algebra ->  Trigonometry-basics -> SOLUTION: The sides of an isosceles triangle have lengths of 10, 10, and 4. Find the measures of the base angles. Round your answers to the nearest tenth. Show the equation used.      Log On


   

Question 1138026: The sides of an isosceles triangle have lengths of 10, 10, and 4. Find the measures of the base angles. Round your answers to the nearest tenth.
Show the equation used.
Answer by MathLover1(20850) About Me  (Show Source):
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let the base angles be alpha and beta=>alpha+=+beta
given: a=10, b=10, and c=4
let altitude to base c be h, then sides a,h, and+c%2F2+form right triangle where a is hypothenuse,+h and+c%2F2 legs
since a=10 and c%2F2=4%2F2=2, we have
h=sqrt%2810%5E2-%282%29%5E2%29
h=sqrt%28100-4%29
h=sqrt%2896%29

using the sine rule:
sin%28alpha%29=h%2F10
sin%28alpha%29=sqrt%2896%29%2F10
alpha=sin%5E-1%28sqrt%2896%29%2F10%29
alpha=78.5°
then
beta=78.5°
and third angle gamma=180-%2878.5%2B78.5%29°=>gamma=180-157+°=>gamma=23°

or, you can do it using the cosine rule:
a=10, b=10, and+c=4
a%5E2=c%5E2%2Bb%5E2-2cb%2Acos%28alpha%29

2cb%2Acos%28alpha%29+=c%5E2%2Bb%5E2-a%5E2
cos%28alpha%29+=%28c%5E2%2Bb%5E2-a%5E2%29%2F%282cb%29
cos%28alpha%29+=%284%5E2%2B10%5E2-10%5E2%29%2F%282%2A4%2A10%29
cos%28alpha%29+=16%2F80
cos%28alpha%29+=1%2F5
alpha=cos%5E-1%281%2F5%29
alpha=78.5° => beta=78.5°