PQRS is a cyclic quadrilateral. Find x + y
50
60
15
0
Correct answer is D
∠PQR + ∠PSR = 180o (opp. angles of cyclic quad. are supplementary)
⇒ 5x - y + 10 + (-2x + 3y + 145) = 180
⇒ 5x - y + 10 - 2x + 3y + 145 = 180
⇒ 3x + 2y + 155 = 180
⇒ 3x + 2y = 180 - 155
⇒ 3x + 2y = 25 ----- (i)
∠QPS + ∠QRS = 180o (opp. angles of cyclic quad. are supplementary)
⇒ -4x - 7y + 150 + (2x + 8y + 105) = 180
⇒ -4x - 7y + 75 + 2x + 8y + 180 = 180
⇒ -2x + y + 255 = 180
⇒ -2x + y = 180 - 255
⇒ -2x + y = -75 ------- (ii)
⇒ y = -75 + 2x -------- (iii)
Substitute (-75 + 2x) for y in equation (i)
⇒ 3x + 2(-75 + 2x) = 25
⇒ 3x - 150 + 4x = 25
⇒ 7x = 25 + 150
⇒ 7x = 175
⇒ x=1757=25
From equation (iii)
⇒ y = -75 + 2(25) = -75 + 50
⇒ y = -25
∴ x + y = 25 + (-25) = 0
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