Given an Isosceles Triangle with only known angles and area, how to solve for a side?

Question

The angles are 116, 32, 32. The area is 360

Answer 1

Isos area= bh/2

Divide in half. The outer side is labeled "c". The mid side is "b". "a" is the short one.

The formula for sides is a^2+b^2=c^2. For the angles: a/sinα=b/sinβ=c/sinγ.

The angles are placed opposite sides.

Since we're talking about a rt triangle; α=32°, β=58°, γ=90°.

Area=ab/2=180.

To figure any triangle; at least 1 side is needed; eveen a formula.

Answer 2

Let the base =b (opposite side of 116°) and height h.

Cut the isosceles triangle by half to make a right triangle, then express b and h in terms of x .

cos 32 = (b/2)/x, so b=2x cos 32

sin 32 = h/x, so h=x sin 32

A=1/2 bh = 1/2⋅2xcos32⋅xsin32 = 360

x² cos32⋅xsin32 = 360.

x=√ 360 / cos32⋅xsin32