- In figure 7.43, A is the centre of the circle. ∠ ABC = 45° and AC = 7√2 cm. Find the…
- In the figure 7.44, O is the centre of the circle. M (arc PQR) = 60° OP = 10 cm. Find…
- In the figure 7.45, if A is the centre of the circle. ∠ PAR = 30°, AP = 7.5, find the…
- In the figure 7.46, if O is the centre of the circle, PQ is a chord. ∠ POQ = 90°, area…
- A chord PQ of a circle with radius 15 cm subtends an angle of 60° with the centre of…
Practice Set 7.4
Question 1.In figure 7.43, A is the centre of the circle. ∠ ABC = 45° and AC = 7√2 cm. Find the area of segment BXC.
Answer:
From the property, we know that, If two sides of a triangle are equal then their corresponding angles are also equal.
So, as AB = AC,
⇒ ∠ ABC = ∠ ACB = 45°
As the sum of angles of a triangle is equal to 180°
⇒∠ ABC + ∠ ACB + ∠ BAC = 180°⇒45° + 45° + ∠ BAC = 180°
⇒90° + ∠ BAC = 180°
⇒ ∠ BAC = 90°
⇒ AT = 49 sq.cm
Area of the sector = one-fourth of a circle
⇒ AS = 77 sq.cm
Area of the shaded region, AR = AS - AT
⇒ AR = 77 – 49
⇒ AR = 28 sq.cm
∴ The area of the shaded region is 28 sq. cm.
Question 2.
In the figure 7.44, O is the centre of the circle. M (arc PQR) = 60° OP = 10 cm.
Find the area of the shaded region. (π = 3.14, √3 = 1.73)
Answer:
Since the angle subtended at centre is 60°
And by the property, if two sides of a triangle are equal then their corresponding angles are also equal.
⇒ ∠ ORP = ∠ OPR
As the sum of all internal angles of a triangle is equal to 180°
⇒ ∠ ORP = ∠ OPR = 60°
⇒ Δ OPR is an equilateral triangle.
⇒ AT = 43.25 sq. cm
Area Of Sector (O-PQR), AS is given as:
⇒ AS = 52.33 sq.cm
Area of shaded region, AR = As – AT
⇒ AR = 52.33 – 43.25
⇒ AR = 9.08 sq.cm
∴ Area of shaded region is 9.08 sq.cm
Question 3.
In the figure 7.45, if A is the centre of the circle. ∠ PAR = 30°, AP = 7.5, find the area of the segment PQR. (π = 3.14)
Answer:
Radius of circle, r = 7.5 cm
∠ PAR = θ = 30°
Area(A – PQR), AS:
⇒ AS = 14.71 sq.cm
Also,
⇒ AT = 14.06 sq. cm
Area of segment PQR, AR = AS - AT
⇒ AR = 14.71 – 14.06 = 0.6562 sq.cm
∴ Area of shaded region is 0.6562 sq.cm
Question 4.
In the figure 7.46, if O is the centre of the circle, PQ is a chord. ∠ POQ = 90°, area of shaded region is 114 cm2, find the radius of the circle. (π = 3.14)
Answer:
Area Of shaded region, AR = 114 sq.cm
Area of sector (O-PRQ),AS = one-fourth of area of circle
Area of Shaded Region, AR = AS – AT
⇒ r = 20 cm
∴ Radius of the circle is 20 cm
Question 5.
A chord PQ of a circle with radius 15 cm subtends an angle of 60° with the centre of the circle. Find the area of the minor as well as the major segment. (π = 3.14)
Answer:
Radius of circle, r = 15cm
Central angle, θ = 60°
Since the angle subtended at centre is 60°
And by the property, if two sides of a triangle are equal then their corresponding angles are also equal.
⇒ ∠ OQP = ∠ OPQ
As the sum of all internal angles of a triangle is equal to 180°
⇒ ∠ OQP = ∠ OPQ = 60°
⇒ Δ OPQ is an equilateral triangle.
AT = 97.32 sq.cm
⇒ AR = 117.75 – 97.32
⇒ AR = 20.43 sq.cm
Now,
⇒ AS = 706.5 – 20.43
⇒ AS = 686.07 sq.cm
∴ The area of minor segment and major segment is 20.43 sq.cm and 686.07 sq.cm respectively
