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Area of the triangle AOB
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14 Jan 2025, 17:05
14 Jan 2025, 17:05
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Expert Reply
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Question Stats:
50% (01:28) correct
50% (02:57) wrong based on 4 sessions
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Attachment:
GRE Area of the triangle AOB.png [ 45.49 KiB | Viewed 286 times ]
Quantity A |
Quantity B |
Area of the triangle AOB |
Area of the shaded region |
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Area of the triangle AOB
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22 Feb 2025, 13:51
22 Feb 2025, 13:51
Expert Reply
OFFICIAL EXPLANATION
GRE circle radius.png [ 95.04 KiB | Viewed 102 times ]
\(\text { Let the radius of the circle above be ' } r \text { ' }\)
The area of the right triangle $\(A O B\)$ is $\(\frac{1}{2} \times\)$ Base $\(\times\)$ Height $\(=\frac{1}{2} \times A O \times O B=\frac{1}{2} \times r \times r=\frac{1}{2}\left(r^2\right)\)($ We have $\(\mathrm{AO}=\mathrm{OB}=\)$ radii $\(=\mathrm{r}\)$ )
Now, the area of the shaded portion $=$ area of sector $\(\operatorname{COD}=\frac{\theta}{360} \times \pi r^2=\frac{60}{360} \times \frac{22}{7} \times \mathrm{r}^2=\frac{11}{21}\left(\mathrm{r}^2\right)\)$
As $\(\frac{1}{2}\)$ is less than $\(\frac{11}{21}\)$, we get $\(\frac{1}{2}\left(\mathrm{r}^2\right)<\frac{11}{21}\left(\mathrm{r}^2\right)\)$
Hence column $\(B\)$ has higher quantity when compared with column $\(A\)$, so the answer is ( $\(B\)$ ).
Attachment:
GRE circle radius.png [ 95.04 KiB | Viewed 102 times ]
\(\text { Let the radius of the circle above be ' } r \text { ' }\)
The area of the right triangle $\(A O B\)$ is $\(\frac{1}{2} \times\)$ Base $\(\times\)$ Height $\(=\frac{1}{2} \times A O \times O B=\frac{1}{2} \times r \times r=\frac{1}{2}\left(r^2\right)\)($ We have $\(\mathrm{AO}=\mathrm{OB}=\)$ radii $\(=\mathrm{r}\)$ )
Now, the area of the shaded portion $=$ area of sector $\(\operatorname{COD}=\frac{\theta}{360} \times \pi r^2=\frac{60}{360} \times \frac{22}{7} \times \mathrm{r}^2=\frac{11}{21}\left(\mathrm{r}^2\right)\)$
As $\(\frac{1}{2}\)$ is less than $\(\frac{11}{21}\)$, we get $\(\frac{1}{2}\left(\mathrm{r}^2\right)<\frac{11}{21}\left(\mathrm{r}^2\right)\)$
Hence column $\(B\)$ has higher quantity when compared with column $\(A\)$, so the answer is ( $\(B\)$ ).
gmatclubot
Area of the triangle AOB [#permalink]
22 Feb 2025, 13:51
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