Difference between revisions of "2023 AMC 8 Problems/Problem 19"
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==Solution 2== | ==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 <math>3:2</math>, we can set the side lengths as <math>3</math> and <math>2</math>, respectively. So, the sum of the trapezoids is <math>\frac{9\sqrt{3}}{4}-\frac{4\sqrt{3}}{4}=\frac{5}{4}\sqrt{3}. We are also told that the three trapezoids are congruent, thus the area of each of them is < | + | 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 <math>3:2</math>, we can set the side lengths as <math>3</math> and <math>2</math>, respectively. So, the sum of the trapezoids is <math>\frac{9\sqrt{3}}{4}-\frac{4\sqrt{3}}{4}=\frac{5}{4}\sqrt{3}</math>. We are also told that the three trapezoids are congruent, thus the area of each of them is <math>\frac{1}{3} \cdot \frac{5}{4}\sqrt{3}=\frac{5}{12}\sqrt{3}</math>. Hence, the area is \frac{\frac{5}{12}\sqrt{3}}{\sqrt{3}}=\boxed{\textbf{(C)}\ \frac{5}{12}}$. |
~MrThinker | ~MrThinker | ||
Revision as of 21:09, 24 January 2023
Contents
Solution 1
By AA~ similarity triangle we can find the ratio of the area of big: small —> 
then there are a relative 
for the 
trapezoids combines. For 
trapezoid it is a relative 
so now the ratio is 
which can simplify to 
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat
Solution 2
Subtracting the larger equilateral triangle from the smaller one yields the sum of the three trapezoids. Since the ratio of the side lengths of the larger to the smaller one is 
, we can set the side lengths as and 
, respectively. So, the sum of the trapezoids is 
. We are also told that the three trapezoids are congruent, thus the area of each of them is 
. Hence, the area is \frac{\frac{5}{12}\sqrt{3}}{\sqrt{3}}=\boxed{\textbf{(C)}\ \frac{5}{12}}$.
~MrThinker
Video Solution by OmegaLearn (Using Similar Triangles)
Animated Video Solution
~Star League (https://starleague.us)
