GRE Prep Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Square $P Q R S$ is tilted $90^{\circ}$ anticlockwise around the point
[#permalink]
09 Aug 2025, 01:20
09 Aug 2025, 01:20
Expert Reply
00:00
Question Stats:
0% (00:00) correct
0% (00:00) wrong based on 0 sessions
Hide Show timer Statistics
Square $\(P Q R S\)$ is tilted $90^{\circ}$ anticlockwise around the point $P$, so that points $Q, R$ and $S$ reach to the points $\(Q^{\prime}, R^{\prime}\)$ and $\(S^{\prime}\)$, respectively. What is the distance covered by the point R if the length of $\(P Q\)$ is 2 ?
(A) $\(\frac{\pi \sqrt{2}}{3}\)$
(B) $\(\frac{\pi \sqrt{2}}{2}\)$
(C) $\(\pi \sqrt{2}\)$
(D) $\(\pi \sqrt{3}\)$
(E) $\(\quad 2 \pi \sqrt{3}\)$
(A) $\(\frac{\pi \sqrt{2}}{3}\)$
(B) $\(\frac{\pi \sqrt{2}}{2}\)$
(C) $\(\pi \sqrt{2}\)$
(D) $\(\pi \sqrt{3}\)$
(E) $\(\quad 2 \pi \sqrt{3}\)$
Re: Square $P Q R S$ is tilted $90^{\circ}$ anticlockwise around the point
[#permalink]
30 Aug 2025, 05:47
30 Aug 2025, 05:47
Expert Reply
Let's analyze the problem step-by-step:
- Square $P Q R S$ has side length $P Q=2$.
- The square is rotated $\(90^{\circ}\)$ anticlockwise around point $P$.
- Points $Q, R, S$ move to new points $\(Q^{\prime}, R^{\prime}, S^{\prime}\)$, respectively.
- We need to find the distance covered by point $R$ during the rotation.
Since the square side length is 2 , the coordinates of the points assuming $P$ as origin could be approximated as:
- $P=(0,0)$
- $Q=(2,0)$
- $R=(2,2)$
- $S=(0,2)$
When the square rotates about $P$ by 90 degrees anticlockwise, the point $R$ moves along a circular arc with center $P$.
The radius of this circular path is the distance from $P$ to $R$ :
$$
\(P R=\sqrt{(2-0)^2+(2-0)^2}=\sqrt{4+4}=\sqrt{8}=2 \sqrt{2}\)
$$
The arc length covered by $R$ when it rotates 90 degrees (which is $\frac{\pi}{2}$ radians) along this circle is:
$$
\(\text { Arc length }=\text { radius } \times \text { angle }=2 \sqrt{2} \times \frac{\pi}{2}=\pi \sqrt{2}\)
$$
So, the distance covered by the point $R$ is $\(\pi \sqrt{2}\)$.
The correct answer is (C) $\(\pi \sqrt{2}\)$.
- Square $P Q R S$ has side length $P Q=2$.
- The square is rotated $\(90^{\circ}\)$ anticlockwise around point $P$.
- Points $Q, R, S$ move to new points $\(Q^{\prime}, R^{\prime}, S^{\prime}\)$, respectively.
- We need to find the distance covered by point $R$ during the rotation.
Since the square side length is 2 , the coordinates of the points assuming $P$ as origin could be approximated as:
- $P=(0,0)$
- $Q=(2,0)$
- $R=(2,2)$
- $S=(0,2)$
When the square rotates about $P$ by 90 degrees anticlockwise, the point $R$ moves along a circular arc with center $P$.
The radius of this circular path is the distance from $P$ to $R$ :
$$
\(P R=\sqrt{(2-0)^2+(2-0)^2}=\sqrt{4+4}=\sqrt{8}=2 \sqrt{2}\)
$$
The arc length covered by $R$ when it rotates 90 degrees (which is $\frac{\pi}{2}$ radians) along this circle is:
$$
\(\text { Arc length }=\text { radius } \times \text { angle }=2 \sqrt{2} \times \frac{\pi}{2}=\pi \sqrt{2}\)
$$
So, the distance covered by the point $R$ is $\(\pi \sqrt{2}\)$.
The correct answer is (C) $\(\pi \sqrt{2}\)$.
gmatclubot
Re: Square $P Q R S$ is tilted $90^{\circ}$ anticlockwise around the point [#permalink]
30 Aug 2025, 05:47
Moderators:
