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Square PQRS is tilted 90 degrees anticlockwise direction around the po

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Math Expert
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Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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06 Oct 2019, 21:33
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Difficulty:

35% (medium)

Question Stats:

56% (01:49) correct 44% (02:58) wrong based on 27 sessions

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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. $$2* \pi \sqrt{3}$$

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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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Updated on: 07 Oct 2019, 03:06
1
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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Originally posted by Dillesh4096 on 06 Oct 2019, 22:09.
Last edited by Dillesh4096 on 07 Oct 2019, 03:06, edited 2 times in total.
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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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07 Oct 2019, 00:20
1
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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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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07 Oct 2019, 01:21
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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

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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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07 Oct 2019, 04:36

90º Rotation anticlockwise direction around the point P is inscribed in a circle.
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Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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Updated on: 08 Oct 2019, 03:46
Quote:
Square PQRS is tilted 90 degrees anti-clockwise 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. (π√2)/3
B. (π√2)/2
C. π√2
D. π√3
E. 2π√3

"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

Originally posted by exc4libur on 07 Oct 2019, 05:50.
Last edited by exc4libur on 08 Oct 2019, 03:46, edited 1 time in total.
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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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07 Oct 2019, 11:48
1
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.

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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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07 Oct 2019, 12:20
We can calculate the r =2sqrt2 and then we can calculate the area of the arc in the sector

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Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po  [#permalink]

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08 Oct 2019, 03:28
Dillesh4096 wrote:
—> RR’ = θ/360*2*π*r = 90/360*2*π*$$\sqrt{2}$$ = π*$$\sqrt{2}$$

90/360*2*π*$$\sqrt{2}$$…
1/4*2*π*$$\sqrt{2}$$…
1/2*π*$$\sqrt{2}$$…
π*$$\sqrt{2}$$/2…
Re: Square PQRS is tilted 90 degrees anticlockwise direction around the po   [#permalink] 08 Oct 2019, 03:28
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