Question

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 `8` and the triangle has an area of `64`. What are the lengths of sides A and B?

Answer 1

Side `color(brown)(a = b = 16.49` units

Explanation:

A median if triangle divides it in 2 triangles of equal areas. And if it is isosceles triangle the median is perpendicular bisector.

`c = 8`

`a = b, height = h`

Half base = c / 2 = 8 / 2 = 4

Area of triangle = 64 sq units

Area of half triangle = 32 sq units

1/2 * 4 * h = 32

`height` `h = 16` units

By using Pythagoras theorem

`a = b = sqrt(4^2 + 16^2) = sqrt 272 = color(brown)(16.49` units

Answer 2

`A = B = 4sqrt 17 "units"`

Explanation:

Let `h` the height#

`"area of triangle" = 1/2 xxh xxC`
`64 = 1/2 xx h xx8`
`h = 16 " units"`

`∆ MHN:`
`MN^2 = h^2 + NH^2`
`A^2 = 16^2 + (8/2)^2`
`A = sqrt 272`
`A = 4sqrt 17`

`A = B = 4sqrt 17 "units"`