ABCD is a rectangle, in which a square PBRQ is inscribed, area of PBRQ is 64 cm2, CP = 1 cm, AR = 4 cm, then find the area of shaded region.?

Question

ABCD is a rectangle, in which a square PBRQ is inscribed, area of PBRQ is 64 cm2, CP = 1 cm, AR = 4 cm, then find the area of shaded region.?

50/3 cm²
70/3 cm²
40/3 cm²
80/3 cm²

LohaHridoy

4 views

2025-01-04 04:42

1 Answers

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AhmedNazir · Accepted Answer · 2022-09-04T04:29:57+06:00

Option 2 : 70/3 cm2

⇒ Area of PBRQ is a square = 64

⇒ Side² = 64

⇒ Side = 8 cm

⇒ PB = BR = RQ = QP = 8 cm

Now, in rectangle ABCD

⇒ CD = AB = AR + BR = 4 + 8 = 12 cm

⇒ AD = BC = CP + PB = 1 + 8 = 9 cm

In ΔARS and ΔCBA,

⇒ ∠A is common,

⇒ ∠R = ∠B = 90°

So, ΔARS and ΔCBA are similar,

⇒ AR ∶ AB = 4 ∶ 12 = 1 ∶ 3

⇒ SR ∶ CB = 1 ∶ 3

⇒ SR = (1/3) × 9 = 3 cm

⇒ QS = 5 cm

In ΔARS and ΔTQS,

⇒ ∠ASR = ∠TSQ (∵ vertically opposite angle)

⇒ ∠R = ∠Q = 90°

So, ΔARS and ΔTQS are similar

⇒ RS ∶ QS = 3 ∶ 5

⇒ AR ∶ QT = 3 ∶ 5

⇒ QT = (5/3) × 4 = 20/3 cm

⇒ TP = PQ – QT = 8 – (20/3) = 4/3 cm

⇒ Area of shaded region = Area of ΔARS + Area of ΔQST + Area of ΔTPC

⇒ Area of shaded region = {(1/2) × 3 × 4} + {(1/2) × 5 × (20/3)} + {(1/2) × (4/3) × 1} = 6 + (50/3) + (2/3) = 70/3

∴ Area of shaded region is (70/3) cm²