An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of 16 16 and the triangle has an area of 40 40. What are the lengths of sides A and B?

2 Answers
Apr 25, 2018

a = b = sqrt{89} a=b=89

Explanation:

Small letters for triangle sides please, dearies. c=16, a=bc=16,a=b.

My favorite formula for the area of a triangle AA with sides a,b,ca,b,c is

16A^2 = 4a^2 c^2 - (b^2 - a^2 - c^2)^2 16A2=4a2c2(b2a2c2)2

Let a=ba=b for an isosceles triangle.

16 A^2 =4a^2 c^2 - c^4 16A2=4a2c2c4

a^2 = {16A^2 + c^4}/{4 c^2} a2=16A2+c44c2

a^2 = {16(40)^2 + 16^4}/{4(16^2)} = 89 a2=16(40)2+1644(162)=89

a = b = sqrt{89} a=b=89

Check:

The altitude hh splits an isosceles triangle with common side aa and base cc is two right triangles, (c/2)^2 + h^2 = a^2(c2)2+h2=a2 or h = sqrt{a^2 - c^2/4}h=a2c24 and an area of

text{area} = 1/2 ch = c/2 \ sqrt{a^2 - c^2/4}

= 1/2 (16) sqrt{ 89 - 64} = 8 sqrt{ 25} = 40 quad sqrt

Lengths of sides A and B
A = 9.4339 or sqrt(89)
B = 9.4339 or sqrt(89)

Explanation:

Isosceles Triangle means two sides are equal.
The area of triangle is
Area = (bxxh)/2
The given are base and the area.
first determine the height before finding A and B.
Substitute Area = 40 and Base C = 16
to the formula
Area = (bxxh)/2
40=(16xxh)/2

or (40) (2) = 16 h
or 80 = 16 h
divide both side by 16
to get the h (height)
80/16 = (16 h)/16
h = 5
since you have the heightenter image source here
cut into two congruent lengths by the height, since we are dealing with an Isosceles triangle, the entire base measures 16, half of the base measures 8.
Now we have two Right Triangles
using Pythagorean Theorem
a^2 + b^2 = c^2
8^2 + 5^2 = c^2
64 + 25 = c^2
89 = c^2
sqrt(89) = C
89 is a prime number
sqrt(89)= 9.4339
A = 9.4339
B = 9.4339