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A painter paints the four equilaterally triangular regions adjoining
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06 May 2025, 02:16
06 May 2025, 02:16
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Question Stats:
60% (01:58) correct
40% (01:39) wrong based on 5 sessions
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A painter paints the four equilaterally triangular regions adjoining the sides of a square $\(A B C D\)$ of diagonal $\(10 \sqrt{2}\)$. What is the ratio of the shaded region to the unshaded region?
GRE figure geometry.png [ 8.44 KiB | Viewed 504 times ]
(A) $\(\sqrt{3}: 2\)$
(B) $\(\sqrt{3}: 1\)$
(C) $\(2: 1\)$
(D) $\(\sqrt{5}: 1\)$
(E) $\(\sqrt{5}: 2\)$
Attachment:
GRE figure geometry.png [ 8.44 KiB | Viewed 504 times ]
(A) $\(\sqrt{3}: 2\)$
(B) $\(\sqrt{3}: 1\)$
(C) $\(2: 1\)$
(D) $\(\sqrt{5}: 1\)$
(E) $\(\sqrt{5}: 2\)$
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Re: A painter paints the four equilaterally triangular regions adjoining
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31 May 2025, 00:57
31 May 2025, 00:57
Expert Reply
\(\begin{aligned}
&\text { Since the diagonal of the square }=10 \sqrt{2} \Rightarrow \text { Side }=10\\
&\text { Shaded region }=4 \cdot\left(\frac{\sqrt{3}}{4} S^2\right)=4 \times \frac{1}{4} \times \sqrt{3} \times 10^2=10^2 \cdot \sqrt{3}
\end{aligned}\)
The area of un-shaded region $\(=10^2\)$
Therefore, the ratio of shaded to un-shaded region $\(=\sqrt{3}: 1\)$.
Thus, the correct answer is $B$.
&\text { Since the diagonal of the square }=10 \sqrt{2} \Rightarrow \text { Side }=10\\
&\text { Shaded region }=4 \cdot\left(\frac{\sqrt{3}}{4} S^2\right)=4 \times \frac{1}{4} \times \sqrt{3} \times 10^2=10^2 \cdot \sqrt{3}
\end{aligned}\)
The area of un-shaded region $\(=10^2\)$
Therefore, the ratio of shaded to un-shaded region $\(=\sqrt{3}: 1\)$.
Thus, the correct answer is $B$.
gmatclubot
Re: A painter paints the four equilaterally triangular regions adjoining [#permalink]
31 May 2025, 00:57
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