hazelnut wrote:

If the hypotenuse of isosceles right triangle ABC has the same length as the height of equilateral triangle DEF, what is the ratio of a leg of triangle ABC to a side of triangle DEF?

A. \(\sqrt{2} / 2\)

B. \(\sqrt{3} / 2\)

C. \(\sqrt{3} /[ 2 *\sqrt{2}]\)

D. \(\sqrt{2} / \sqrt{3}\)

E. \(\frac{3}{2}\)

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Isosceles right triangleAn 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 triangleAny altitude of an equilateral triangle creates two congruent triangles with angle measures 30-60-90Side 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

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