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MATH / An equilateral triangle has one vertex on each of the sides of the right…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

An equilateral triangle has one vertex on each of the sides of the right triangle with side lengths 232\sqrt3, 55, and 37\sqrt{37}, as shown. Find the smallest possible area of the equilateral triangle. [asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy]
Plain-text mathematical notation (without MathML)
An equilateral triangle has one vertex on each of the sides of the right triangle with side lengths 2√(3), 5, and √(37), as shown.  Find the smallest possible area of the equilateral triangle.

[asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy]
Original LaTeX notation
An equilateral triangle has one vertex on each of the sides of the right triangle with side lengths $2\sqrt3$, $5$, and $\sqrt{37}$, as shown.  Find the smallest possible area of the equilateral triangle.

[asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy]

level

Level 4

type

Precalculus

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Source and history

task_id
0ad9e8d7-d1cf-5e83-9293-08ed4f0916cb
task_key
precalculus--test--0ad9e8d7-d1cf-5e83-9293-08ed4f0916cb
task_revision_id
1
config / split
precalculus / test

Official source

initial import