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24 Nov 2021, 02:30
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P, Q, S, and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triangle and PS is a diameter of the circle.
What is the Perimeter of the quadrilateral PQSR?
A. \(2r(1+\sqrt{3})\)
B. \(2r(2+\sqrt{3})\)
C. \(r(1+\sqrt{5})\)
D. \(2r(1+\sqrt{5})\)
E. \(2r+\sqrt{3}\)
What is the Perimeter of the quadrilateral PQSR?
A. \(2r(1+\sqrt{3})\)
B. \(2r(2+\sqrt{3})\)
C. \(r(1+\sqrt{5})\)
D. \(2r(1+\sqrt{5})\)
E. \(2r+\sqrt{3}\)
24 Nov 2021, 05:20
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Sides of quad will be\( r, \sqrt{3}r, r, \sqrt{3}r\) .
A is the correct answer
A is the correct answer
Bunuel
P, Q, S, and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triangle and PS is a diameter of the circle.
What is the Perimeter of the quadrilateral PQSR?
A. \(2r(1+\sqrt{3})\)
B. \(2r(2+\sqrt{3})\)
C. \(r(1+\sqrt{5})\)
D. \(2r(1+\sqrt{5})\)
E. \(2r+\sqrt{3}\)
What is the Perimeter of the quadrilateral PQSR?
A. \(2r(1+\sqrt{3})\)
B. \(2r(2+\sqrt{3})\)
C. \(r(1+\sqrt{5})\)
D. \(2r(1+\sqrt{5})\)
E. \(2r+\sqrt{3}\)
31 Dec 2021, 03:51
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Please refer to attached figure.
Diameter PS divides the Quadrilateral PQSR into two SIMILAR RIGHT TRIANGLES
Since, the triangle PQR is an equilateral triangle, therefore angle subtended by each arc: QP = PR = SQ = 120
△QOP is an Isosceles Triangle with sides OP = OQ = r and ∠QOP = 120, therefore, ∠QPO = ∠OQP = 30
△PQS is a Right Triangle with ∠PQS = 90 because the chord PS (Diameter) subtends 180 degree at the center therefore angle subtended in the remaining arc will be half of it.
∴ In △PQS, ∠QPS = 30, ∠PQS = 90 and ∠QSP = 60
We know, ratio of sides in a 30:60:90 = 1: √3 : 2
∠PQS = 90 = 2r
∠QSP = 60 = √3/2*2r = √3r
∠QPS = 30 = 1/2*2r = r
∴ Perimeter of Quadrilateral PQSR = 2(√3r) + 2(r) = 2r(√3+1) (A)
Diameter PS divides the Quadrilateral PQSR into two SIMILAR RIGHT TRIANGLES
Since, the triangle PQR is an equilateral triangle, therefore angle subtended by each arc: QP = PR = SQ = 120
△QOP is an Isosceles Triangle with sides OP = OQ = r and ∠QOP = 120, therefore, ∠QPO = ∠OQP = 30
△PQS is a Right Triangle with ∠PQS = 90 because the chord PS (Diameter) subtends 180 degree at the center therefore angle subtended in the remaining arc will be half of it.
∴ In △PQS, ∠QPS = 30, ∠PQS = 90 and ∠QSP = 60
We know, ratio of sides in a 30:60:90 = 1: √3 : 2
∠PQS = 90 = 2r
∠QSP = 60 = √3/2*2r = √3r
∠QPS = 30 = 1/2*2r = r
∴ Perimeter of Quadrilateral PQSR = 2(√3r) + 2(r) = 2r(√3+1) (A)
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