Question

What are the dimensions of the largest 30-60-90 triangle that will fit inside a 45-45-90 triangle with leg length 14in.? Sketch your solution

Answer 1

The dimensions would be 7√2, 7√6 and 14√2.

A 45-45-90 triangle has two equal legs and a hypotenuse that is equal to the leg multiplied by the square root of two (t, t, t√2).

A 30-60-90 triangle has legs that are equal to t, t√3, and 2t.  

The longest side on the 45-45-90 triangle will be 14√2.  This will also be the longest side of the 30-60-90 triangle; this gives us

2t = 14√2

Dividing both sides by 2, we have
2t/2 = 14√2/2
t = 7√2

We now have t and 2t; we lack t√3:

t = 7√2
t√3 = 7√2(√3) = 7√6.