2014 AMC 12A Problems/Problem 10

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Problem

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

$\textbf{(A) }\dfrac{\sqrt3}4\qquad \textbf{(B) }\dfrac{\sqrt3}3\qquad \textbf{(C) }\dfrac23\qquad \textbf{(D) }\dfrac{\sqrt2}2\qquad \textbf{(E) }\dfrac{\sqrt3}2$

Solution 1

Reflect each of the triangles over its respective side. Then since the areas of the triangles total to the area of the equilateral triangle, it can be seen that the triangles fill up the equilateral one and the vertices of these triangles concur at the circumcenter of the equilateral triangle. Hence the desired answer is just its circumradius, or $\boxed{\dfrac{\sqrt3}3\textbf{ (B)}}$.

(Solution by djmathman)

Solution 2

Since the total area of each congruent isosceles triangle is the same, the area of each is ${\dfrac{1}3$ (Error compiling LaTeX. ! Missing } inserted.) the total area of the equilateral triangle of side length 1, or ${\dfrac{1}3$ (Error compiling LaTeX. ! Missing } inserted.) x ${\dfrac{\sqrt3}4$ (Error compiling LaTeX. ! Missing } inserted.). Likewise, the area of each can be defined as ${\dfrac{bh}2$ (Error compiling LaTeX. ! Missing } inserted.) with base $b$ equaling 1, meaning that ${\dfrac{h}2$ (Error compiling LaTeX. ! Missing } inserted.) = ${\dfrac{1}3$ (Error compiling LaTeX. ! Missing } inserted.) x ${\dfrac{\sqrt3}4$ (Error compiling LaTeX. ! Missing } inserted.), or $h$ = ${\dfrac{\sqrt3}6$ (Error compiling LaTeX. ! Missing } inserted.). A side length of the isosceles triangle is the hypotenuse with legs ${\dfrac{b}2$ (Error compiling LaTeX. ! Missing } inserted.) and $h$. Using the Pythagorean Theorem, the side length is $\sqrt{{(\dfrac{1}2)}^2 + {({\dfrac{\sqrt3}6)}^2}$ (Error compiling LaTeX. ! Missing } inserted.), or $\boxed{\dfrac{\sqrt3}3\textbf{ (B)}}$.

(Solution by johnstucky)

See Also

2014 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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