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.
Solution:
We use the concepts of areas of circles and areas of triangles to solve the problem.
AB and CD are diameters of the circle with center O
∴ OD = OC = OA = OB = Radius of the circle R = 7 cm
∴ AB = 2R = 14 cm
Radius of shaded circular region, r = OD/2 = 7/2 cm
Area of the shaded smaller circular region = πr²
= π (7/2 cm)2
= 22/7 × 7/2 × 7/2 cm2
= 77/2 cm2
= 38.5 cm2
Area of the shaded segment of larger circular region = Area of semicircle ACB - Area of ΔABC
= 1/2 π(OA)2 - 1/2 × AB × OC
= 1/2 πR2 - 1/2 × 2R × R
= 1/2 × 22/7 × 72 - 1/2 × 14 × 7
= 77 - 49
= 28 cm2
Area of the shaded region = Area of the shaded smaller circular region + Area of the shaded segment of larger circular region
= 38.5 cm2 + 28 cm2
= 66.5 cm2
☛ Check: NCERT Solutions Class 10 Maths Chapter 12
Video Solution:
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
NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.3 Question 9
Summary:
The area of the shaded region if 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 with OA = 7 cm is 66.5 cm2.
☛ Related Questions:
- The area of an equilateral triangle ABC is 17320.5 cm2 . With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Fig. 12.28). Find the area of the shaded region. (Use π = 3.14 and √3 = 1.73205)
- On a square handkerchief, nine circular designs each of radius 7 cm are made (see Fig. 12.29). Find the area of the remaining portion of the handkerchief.
- In Fig. 12.30, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the(i) quadrant OACB,(ii) shaded region
- In Fig. 12.31, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14).