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.

The triangle lengths would be 14√2, 7√2 and 7√6.

The sides of a 45-45-90 triangle can be listed as t, t and t√2.  The legs are each 14, so the hypotenuse will be 14√2.

This will also be the longest side of the 30-60-90 triangle.  Its sides can be listed as t, 2t and t√3.  Since the hypotenuse is the longest side, we have:

2t=14√2

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

The remaining side will be t√3:

t = 7√2
t√3 = 7√2(√3) = 7√6

1+1=22×1=22-1=11-1=1duh