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

Answer 1

`abs(A)=abs(B)=sqrt(3^2+(64/3)^2)~~21.5434`

Explanation:

Using `abs(C)=6` as the base and `64` as the area
the height of the triangle (relative to `C`) is
`color(white)("XXX")h=64/3` (Since `"Area" = 1/2*base*h`)

Since `triangleABC` is isosceles with `abs(A)=abs(B)`
the height (relative to side `C`) bisects the length of `C` (innto two segments of `3` each).

Using the Pythagorean Theorem
`color(white)("XXX")abs(A)=sqrt(3^2+h^2)=sqrt(3^2+(64/3)^2)`

Using a calculator or computer we can evaluate this as
`color(white)("XXX")abs(A)~~21.5434`

(and `abs(B)=abs(A)`)