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The two semi - circle are drawn as shown in the figure. Chord CD is of the length 16 cm is parallel to diameter AB of bigger semi - circle and touches the smaller semi - circle. What is the area of shaded region ?

The two semi - circle are drawn as shown in the figure. Chord CD is of the length 16 cm is parallel to diameter AB of bigger semi - circle and touches the smaller semi - circle. What is the area of shaded region ? Correct Answer 32π cm<sup>2</sup>

Given - 

Length of chord CD = 16 cm

Concept:

Theorem:

The perpendicular from the center of a circle to a chord bisects the

chord. 

If a semi-circle has radius r,

Area = (πr2/2)

Solution: 

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Let O be the center of smaller and M be the center of bigger semi - circle and radius are r and R respectively.

⇒ OP = MN = OR = r

⇒ MB = MD = R

⇒ CD = 16 cm

⇒ MN is perpendicular to CD.

We know that,

The perpendicular from the center of a circle to a chord bisects the

chord.

So, ND = 8 cm

⇒ In ΔMND,

⇒ R2 = r2 + 82

⇒ R2 - r2 = 64

⇒ area of the shaded region = (1/2) × π × (R2 - r2)

⇒ area of the shaded region = (1/2) × π × 64 = 32π cm2 

∴ area of the shaded part = 32π cm2

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