Manhattan Prep
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. \(\dfrac{3}{2}\)
OA C
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If the hypotenuse of isosceles right triangle ABC has the same length as the height of equilateral triangle DEF...
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Solution:AAPL wrote: ↑Wed Jan 13, 2021 4:19 pmManhattan Prep
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. \(\dfrac{3}{2}\)
OA C
Let the hypotenuse of isosceles right triangle ABC be 1; thus, the height of equilateral triangle DEF is 1 also. Furthermore, a leg of triangle ABC is 1/√2, and a side of triangle DEF is 1/√3 x 2 = 2/√3. Therefore, the ratio is (1/√2) / (2/√3) = 1/√2 x √3/2 = √3/(2√2).
Answer: C
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