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Geometry – Practice – Rectangle inscribed inside circle!!!

GEO_PRACTICE_Q1 .In the figure below (not drawn to scale), a rectangle ABCD is inscribed in the circle with centre at O. The length of side AB is greater than that of side BC. The ratio of the area of the circle to the area of the rectangle ABCD is π:√3. The line segment DE intersects AB at E such that ∠ ODC = ∠ ADE. What is the ratio AE:AD?

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A. 1:√3

B. 1:√2

C. 1:2√3

D. 1:2

Ans: Let the radius of the circle be ‘R’ and ∠ODC=∠ADE= θ.

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If OM is drawn perpendicular to DC,

DM = R* Cos θ

OM = R* Sin θ

Length of rectangle ABCD,

AB= CD= 2DM = 2R* Cos θ

AD=BC= 2OM=2R* Sin θ

Area of rectangle= AB * BC = 2R* Cos θ * 2R* Sin θ = 2R² Sin 2θ

Area of circle = πR²

According to the question,

πR²:2R² Sin 2θ = π:√3

=> Sin 2θ =√3/2 =Sin 60 deg.

θ =30 deg.

In triangle ADE, tan θ=AE/AD

AE:AD=tan 30 deg = 1:√3. Option A.

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