If the hypotenuse of isosceles right triangle ABC has the same length

Isosceles right triangle
An isosceles right triangle has angle measures 45-45-90
Side lengths opposite those angles are in ratio
\(x: x: x\sqrt{2}\) or
\(1: 1: \sqrt{2}\)

Equilateral triangle
Any altitude of an equilateral triangle creates two congruent triangles with angle measures 30-60-90
Side lengths opposite those angles are in ratio
\(x: x\sqrt{3}: 2x\) or
\(1: \sqrt{3}: 2\)

The hypotenuse of isosceles right ∆ ABC = the height (altitude) of equilateral ∆ DEF
What is the ratio of a leg of ∆ ABC to a side of ∆ DEF?

1) Assign a value for a leg of ∆ ABC. Find hypotenuse BC
Let a leg of the isosceles right triangle, opposite a 45° angle \(= 1 = x\)
Hypotenuse BC, opposite the 90° angle = \(x\sqrt{2} = (1*\sqrt{2}) = \sqrt{2}\)
BC = \(\sqrt{2}\)

2) Find side length of ∆ DEF from BC and 30-60-90 ∆ side ratios
BC = EG
Height EG, opposite the 60° angle = \(x\sqrt{3}\)
Height EG also = \(\sqrt{2}\)

\(x\sqrt{3} = \sqrt{2}\)

\(x = \frac{\sqrt{2}}{\sqrt{3}}\)

Side of ∆ DEF = \((2x) = (2 *\frac{\sqrt{2}}{\sqrt{3}})= \frac{2\sqrt{2}}{\sqrt{3}}\)

3) Ratio of a leg of ∆ ABC to a side of ∆ DEF?

Leg of ∆ ABC = 1 (from above)

Side of ∆ DEF = \(\frac{2\sqrt{2}}{\sqrt{3}}\)

\(\frac{1}{(\frac{2√2}{√3})} = \frac{√3}{2√2}\)

ANSWER C