2023 AMC 8 Problems/Problem 19
Contents
Problem
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 
the side length of the larger triangle. What is the ratio of one trapezoid to the area of the inner triangle?
<Need image>
Solution 1
By AA~ similarity triangle we can find the ratio of the area of big: small —> 
then there are a relative 
for the 
trapezoids combines. For 
trapezoid it is a relative 
so now the ratio is 
which can simplify to 
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat
Solution 2
Subtracting the larger equilateral triangle from the smaller one yields the sum of the three trapezoids. Since the ratio of the side lengths of the larger to the smaller one is 
, we can set the side lengths as and 
, respectively. So, the sum of the trapezoids is 
. We are also told that the three trapezoids are congruent, thus the area of each of them is 
. Hence, the ratio is 
.
~MrThinker
Video Solution by OmegaLearn (Using Similar Triangles)
Animated Video Solution
Video Sution by SpraedTheMathLove using Area-Similarity Relationship
https://www.youtube.com/watch?v=92hAg3JjqZI
~Star League (https://starleague.us)
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
