2018 AMC 12A Problem #8

Geometry Level 2

All of the triangles in the diagram below are similar to iscoceles triangle A B C , ABC, in which A B = A C . AB=AC. Each of the 7 smallest triangles has area 1, and △ A B C \triangle ABC has area 40. What is the area of trapezoid D B C E DBCE ?

Next problem

Previous problem

All problems

22 20 16 24 18

Steven Yuan by Steven Yuan , 20, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Since the area of ABC is 40 times that of a small triangle, the side of a small triangle is 10 20 \frac{\sqrt{10}}{20} of the side of ABC. So the side of DEA is 5 times that factor and thus the area of 2/5th of ABC. This leads to 24 for the area of the trapezoid.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Jerry McKenzie
Feb 9, 2018

Continue drawing the triangles up, you find there are 16 triangles inside Δ A D E \Delta ADE . Since they are all similar, that makes Δ A D E \Delta ADE area 16 16 . Thus the trapezoid is 40 − 16 = 24 40-16=24 .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...