2023 AMC 8 Problem 19 - Math Gold Medalist

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2023 AMC 8

Problem 19

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

$[asy] // Diagram by TheMathGuyd pair A,B,C; A=(0,1); B=(sqrt(3)/2,-1/2); C=-conj(B); fill(2B--3B--3C--2C--cycle,grey); dot(3A); dot(3B); dot(3C); dot(2A); dot(2B); dot(2C); draw(2A--2B--2C--cycle); draw(3A--3B--3C--cycle); draw(2A--3A); draw(2B--3B); draw(2C--3C); [/asy]$

$\textbf{(A) } 1 : 3 \qquad \textbf{(B) } 3 : 8 \qquad \textbf{(C) } 5 : 12 \qquad \textbf{(D) } 7 : 16 \qquad \textbf{(E) } 4 : 9$

Related Ideas

Symmetry

Area

Drawing Perpendiculars

Right Triangle: 30 60 90

Hint

The trapezoid is isosceles.

Draw two altitudes of the trapezoid

Altitudes of trapezoid are opposite to angle 30