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A square is formed by joining midpoints of another square. Perimeter
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12 Jan 2025, 23:30
12 Jan 2025, 23:30
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66% (01:08) wrong based on 3 sessions
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A square is formed by joining midpoints of another square. Perimeter of the larger square is X.
Quantity A |
Quantity B |
Perimeter of smaller square |
\(\frac{ X}{2}\) |
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A square is formed by joining midpoints of another square. Perimeter
[#permalink]
22 Feb 2025, 01:00
22 Feb 2025, 01:00
Expert Reply
OFFICIAL EXPLANATION
\(\text { Let } \mathrm{ABCD} \text { be the square whose midpoints are joined to form another square say } \mathrm{PQRS}\)
GRE ABCD.png [ 47.93 KiB | Viewed 11 times ]
We know that the perimeter of the larger square i.e. $\(A B C D\)$ is $x$, so we get side of the square ABCD as $\(\frac{X}{4}\)$ each (Perimeter of square is $\(=4 \times$ Side )\).
Using Pythagoras theorem i.e. Hypotenuse $\({ }^2=\)$ Perpendicular $\(^2+\)$ Base $\(^2\)$ in triangle APS, we get
$\(\mathrm{PS}=\sqrt{AP^2+AS^2}=\sqrt{ (\frac{x}{8})^2+(\frac{x}{8})^2}=\frac{x \sqrt{2}}{8}\)$ (The length of $\(\mathrm{AS}=\mathrm{AP}=\)$ half of the $\mathrm{AB}=\frac{x}{4}[/m]$ ).
So, the perimeter of the smaller square i.e. $\(\mathrm{PQRS}=4 \times \operatorname{Side}\)\(=4 \times \frac{x\sqrt{2}}{8}=\frac{x\sqrt{2}}{2}=\frac{x}{\sqrt{2}}\)$ Column A quantity) which is clearly greater than $\(\frac{x}{2}\)$ (= column B quantity).
Hence the answer is (A).
\(\text { Let } \mathrm{ABCD} \text { be the square whose midpoints are joined to form another square say } \mathrm{PQRS}\)
Attachment:
GRE ABCD.png [ 47.93 KiB | Viewed 11 times ]
We know that the perimeter of the larger square i.e. $\(A B C D\)$ is $x$, so we get side of the square ABCD as $\(\frac{X}{4}\)$ each (Perimeter of square is $\(=4 \times$ Side )\).
Using Pythagoras theorem i.e. Hypotenuse $\({ }^2=\)$ Perpendicular $\(^2+\)$ Base $\(^2\)$ in triangle APS, we get
$\(\mathrm{PS}=\sqrt{AP^2+AS^2}=\sqrt{ (\frac{x}{8})^2+(\frac{x}{8})^2}=\frac{x \sqrt{2}}{8}\)$ (The length of $\(\mathrm{AS}=\mathrm{AP}=\)$ half of the $\mathrm{AB}=\frac{x}{4}[/m]$ ).
So, the perimeter of the smaller square i.e. $\(\mathrm{PQRS}=4 \times \operatorname{Side}\)\(=4 \times \frac{x\sqrt{2}}{8}=\frac{x\sqrt{2}}{2}=\frac{x}{\sqrt{2}}\)$ Column A quantity) which is clearly greater than $\(\frac{x}{2}\)$ (= column B quantity).
Hence the answer is (A).
gmatclubot
A square is formed by joining midpoints of another square. Perimeter [#permalink]
22 Feb 2025, 01:00
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