Square PQRS is tilted 90 degrees anticlockwise direction around the po

Question

Competition Mode Question

Square PQRS is tilted 90 degrees anticlockwise direction around the point P, so that points Q, R, S reach the points Q', R', S' respectively. What is the distance covered by the point R if the length of PQ is 2.

A.  (\pi * \sqrt{2})/3

B.  (\pi * \sqrt{2})/2

C.  \pi \sqrt{2}

D.  \pi \sqrt{3}

E.  2* \pi \sqrt{3}

CareerGeek · Accepted Answer

Let the point P be origin.
And square be placed in first quadrant with Q = (2,0), R = (2,2) & S = (0,2)

When square is rotated anti-clockwise 90 deg, point R travels along the arc of the circle of radius PR =  \sqrt{2}

The new coordinates are P(0,0), Q’(0,2), R’(-2,2), S’(-2,0)

Distance travelled by point R is nothing but the length of the arc RR’ of circle with radius PR and angle 90 deg at the center

—> RR’ = θ/360*2*π*r = 90/360*2*π* \sqrt{2}  = π* \sqrt{2}

IMO Option C

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madgmat2019 · Answer

If we consider a circular in which square is inscribed,Radius of the circle will be 2√2

Distance covered by R is 2πr/4

=π√2

OA:C

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unraveled · Answer

Square PQRS is tilted 90 degrees anticlockwise direction around the point P, so that points Q, R, S reach the points Q', R', S' respectively. What is the distance covered by the point R if the length of PQ is 2.

Since point P is the hinge point we can take point Q either on left side or on right side of P. As square PQRS is rotated anticlockwise with center at point P, radius would be equal to diagonal PR where PR s equal to √2 * side of square.

PR = 2√2

Here R would cover a distance along the length of perimeter of circle with radius 2√2.
Since only 90 deg is rotated, point R would cover only one - fourth the length of circle's perimeter.

Perimeter of circle with radius PR = 2 * π * PR
= 2π2√2
= 4π√2

Distance covered by point R =  \frac{1}{4} * 4π√2 
= π√2

Answer C.

lacktutor · Answer

Please, Find The Attached

90º Rotation anticlockwise direction around the point P is inscribed in a circle.

exc4libur · Answer

"AROUND POINT P" means that P is the origin; 
therefore, the diagonal of the square PQRS is the radius, not the diameter!

square PQRS side = 2
square PQRS diagonal = circle's radius r = side√2 = (2)√2
circle's circumference = 2πr = 2π(2)√2 = 4π√2
distance covered by R to R' (90 deg…360/90…1/4) = (1/4)•4π√2 = π√2

Answer (C)

eakabuah · Answer

If we treat it as though it is an a square inscribed within a circle, then we know that the radius of the circle within which a square is inscribed is half the diagonal of the square. So we already know that PQ=2, and the diagonal of the square = length of side x √2
hence diagonal = 2*√2
Radius of the circle = half of the diagonal = √2

Now the distance covered by the square when it is rotated 90degrees = circumference of the circle * 90/360 = 2*π*√2 *90/36 
= 2*π*√2 * 1/4 = π*√2 / 2.

The answer is therefore B.

SD0715 · Answer

We can calculate the r =2sqrt2 and then we can calculate the area of the arc in the sector

The answer should be C.

exc4libur · Answer

your calculation is wrong:
90/360*2*π* \sqrt{2} …
1/4*2*π* \sqrt{2} …
1/2*π* \sqrt{2} …
π* \sqrt{2} /2…

lpp15 · Answer

key to this question in my case was to correctly draw the rotation.
Rotation is about Center point P - I initially didn’t read the question correctly and rotated the whole square rather than just Q, R,S
Which resulted in diagonal of the square = radius of the circle rather than my initial view of diagonal being the diameter which was screwing up the equation.

Thanks,
Lasya

bumpbot · Answer

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Square PQRS is tilted 90 degrees anticlockwise direction around the po

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Square PQRS is tilted 90 degrees anticlockwise direction around the po

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