Question

The figure is constructed from $11$ line segments, each of which has length $2.$ The area of pentagon $ABCDE$ can be written as $\sqrt m+\sqrt n,$ where $m$ and $n$ are positive integers. What is $m+n?$

$20$

$21$

$22$

$23$

$24$

Solution:

The eleven equal segments form two rhombi (each two equilateral triangles of side $2$ ) sharing the apex $A,$ with $C$ and $D$ joined by a final segment. The figure is symmetric about the vertical line through $A.$

Placing $A=(0,0)$ at the top, the two bottom vertices come out to $C=(-1,-\sqrt{11})$ and $D=(1,-\sqrt{11}),$ with $B$ and $E$ the outer corners at height $-\tfrac{\sqrt{11}}{2}+\tfrac{1}{2\sqrt3}.$

Applying the shoelace formula to pentagon $ABCDE$ gives area $\sqrt{11}+2\sqrt3=\sqrt{11}+\sqrt{12}.$

So $m+n=11+12=23.$

Thus, the correct answer is D.

2021 AMC 12B Spring Problem 15

Solution:

Problem 15 in Other Years