**Chapter 1 Angle and its Measurement Ex 1.2**

## Chapter 1 Angle and its Measurement Ex 1.2

**Question 1.Find the length of an arc of a circle which subtends an angle of 108Â° at the centre, if the radius of the circle is 15 cm.Solution:**

Here, r = 15cm and

= 9Ï€ cm.

**Question 2.The radius of a circle is 9 cm. Find the length of an arc of this circle which cuts off a chord of length equal to length of radius.Solution:**

Here, r = 9cm

Let the arc AB cut off a chord equal to the radius of the circle.

Since OA = OB = AB,

Î”OAB is an equilateral triangle.

mâˆ AOB = 60Â°

Î¸ = 60Â°

**Question 3.Find the angle in degree subtended at the centre of a circle by an arc whose length is 15 cm, if the radius of the circle is 25 cm.Solution:**

Here, r = 25 cm and S = 15 cm

Since S = r.Î¸,

15 = 25 x Î¸

**Question 4.A pendulum of length 14 cm oscillates through an angle of 18Â°. Find the length of its path.Solution:**

**Question 5.Two arcs of the same length subtend angles of 60Â° and 75Â° at the centres of the two circles. What is the ratio of radii of two circles?Solution:**

Let , and be the radii of the two circles and let their arcs of same length S subtend angles of 60Â° and 75Â° at their centres.

Angle subtended at the centre of the first circle,

Angle subtended at the centre of the second circle,

**Question 6.The area of the circle is 2571 sq.cm. Find the length of its arc subtending an angle of 144Â° at the centre. Also find the area of the corresponding sector.Solution:**

Area of circle = Ï€r

^{2}

But area is given to be 25 Ï€ sq.cm

âˆ´ 25Ï€ = Ï€r

^{2}

âˆ´ r

^{2}= 25

âˆ´ r = 5 cm

**Question 7.OAB is a sector of the circle having centre at O and radius 12 cm. If mâˆ AOB = 45Â°, find the difference between the area of sector OAB and Î”AOB.Solution:**

Here, r = 12 cm

Draw AM âŠ¥ OB

In Î”OAM,

[Note: The question has been modified.]

**Question 8.OPQ is the sector of a circle having centre at O and radius 15 cm. If mâˆ POQ = 30Â°, find the area enclosed by arc PQ and chord PQ.Solution:**

Here, r = 15 cm

mâˆ POQ = 30Â°

\(\left(30 \times \frac{\pi}{180}\right)^{c}[/larex]

âˆ´ Î¸ = [latex]\left(\frac{\pi}{6}\right)^{c}\)

Draw QM âŠ¥ OP

In Î”OQM,

Shaded portion indicates the area enclosed by arc PQ and chord PQ.

âˆ´ A(shaded portion)

= A(sector OPQ) â€“ A(Î”OPQ)

**Question 9.The perimeter of a sector of the circle of area 25Ï€ sq.cm is 20 cm. Find the area of sector.Solution:**

Area of circle = Ï€

But area is given to be 25Ï€ sq.cm.

âˆ´ 25Ï€ = Ï€

âˆ´ = 25

âˆ´ r = 5 cm

Perimeter of sector = 2r + S

But perimeter is given to be 20 cm.

âˆ´ 20 = 2(5) + S

âˆ´ 20 = 10 + S

âˆ´ S = 10 cm

Area of sector =1/2 x r x S

=1/2 x 5 x 10

= 25sq.cm.

**Question 10.The perimeter of a sector of the circle of area 64 7i sq.cm is 56 cm. Find the area of the sector.Solution:**

Area of circle = Ï€

But area is given to be 25Ï€ sq.cm.

âˆ´ 64Ï€ = Ï€

âˆ´ = 64

âˆ´ r = 8 cm

Perimeter of sector = 2r + S

But perimeter is given to be 20 cm.

âˆ´ 56 = 2(5) + S

âˆ´ 56 = 16 + S

âˆ´ S = 40 cm

Area of sector =1/2 x r x S

=1/2 x 8 x 40

= 160sq.cm.