# In Fig. 12.33, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region

**Solution:**

We use the formula for areas of semi-circles and sector of circles to solve the problem.

To find the area of semi-circle, we need to find the radius or diameter (BC) of the semicircle.

ΔABC is a right-angled triangle, right-angled at A (ABC being a quadrant).

AB = AC = 14 cm [Radius of the circle]

Using Pythagoras theorem, we can find the hypotenuse (BC) of ΔABC.

BC^{2} = AB^{2} + AC^{2}

= (14 cm)^{2} + (14 cm)^{2}

BC = √2 × (14 cm)²

= 14√2 cm

∴ Radius of semicircle BDC, r = BC/2 = 14√2/2 cm = 7√2 cm

Area of the shaded region = Area of semicircle - (Area of quadrant ABC - Area ΔABC)

= πr^{2}/2 - [90°/360° × π(14)^{2} - 1/2 × AC × AB]

= π(7√2)^{2}/2 - [π(14)^{2}/4 - 1/2 × 14 × 14]

= [(22 × 7 × 7 × 2)/(7 × 2)] - [(22 × 14 × 14)/(7 × 4) - 7 × 14]

= 154 - (154 - 98)

= 98 cm^{2}

**ā Check: **NCERT Solutions Class 10 Maths Chapter 12

**Video Solution:**

## In Fig. 12.33, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.3 Question 15

**Summary:**

The area of the shaded region where ABC is a quadrant of a circle of radius 14 cm and a semicircle drawn with BC as the diameter is 98 cm^{2}.

**ā Related Questions:**

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- In Fig. 12.25, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region.
- Fig. 12.26 depicts a racing track whose left and right ends are semicircular. Fig. 12.26 The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find :(i) the distance around the track along its inner edge(ii) the area of the track.
- In Fig. 12.27, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

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