**Chapter 1 Angle and its Measurement Miscellaneous Exercise 1**

## Chapter 1 Angle and its Measurement Miscellaneous Exercise 1

**I. Select the correct option from the given alternatives.**

**Question 1.**

(A) 246°

(B) 264°

(C) 224°

(D) 426°

**Answer:**

(B) 264°

**Question 2.156° is equal toAnswer:**

(B)

**Question 3.A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88 metres when it traces the angle of 12° at the centre, then the length of the rope is**

(A) 70 m

(B) 55 m

(C) 40 m

(D) 35 m

**Answer:**

(A) 70 m

**Question 4.A pendulum 14 cm long oscillates through an angle of 12°, then the angle of the path described by its extremities isAnswer:**

(D)

**Question 5.Angle between hands of a clock when it shows the time 9 :45 is**

(A) (7.5)°

(B) (12.5)°

(C) (17.5)°

(D) (22.5)°

**Answer:**

(D) (22.5)°

**Question 6.20 metres of wire is available for fencing off a flower-bed in the form of a circular sector of radius 5 metres, then .the maximum area (in sq. m.) of the flower-bed is**

(A) 15

(B) 20

(C) 25

(D) 30

**Answer:**

(C) 25

r + r + rθ = 20m

2r + rθ = 20

**Question 7.If the angles of a triangle are in the ratio 1:2:3, then the smallest angle in radian is**

**Question 8.A semicircle is divided into two sectors whose angles are in the ratio 4:5. Find the ratio of their areas?**

(A) 5:1

(B) 4:5

(C) 5:4

(D) 3:4

**Answer:**

(B) 4:5

**Question 9.Find the measure of the angle between hour- hand and the minute hand of a clock at twenty minutes past two.**

(A) 50°

(B) 60°

(C) 54°

(D) 65°

**Answer:**

(A) 50°

**Question 10.The central angle of a sector of circle of area 9π sq.cm is 60°, the perimeter of the sector is**

(A) π

(B) 3 + π

(C) 6 + π

(D) 6

**Answer:**

(C) 6 + π

**II. Answer the following.**

**Question 1.**

∴ Number of sides of a regular polygon = 8.

**Question 2.Two circles each of radius 7 cm, intersect each other. The distance between their centres is 7√2 cm. Find the area common to both the circles.Solution:**

**Question 3.∆PQR is an equilateral triangle with side 18 cm. A circle is drawn on segment QR as diameter. Find the length of the arc of this circle within the triangle.Solution:**

Let ‘O’ be the centre of the circle drawn on QR as a diameter.

Let the circle intersect seg PQ and seg PR at points M and N respectively.

Since l(OQ) = l(OM),

m∠OM Q = m∠OQM = 60°

m∠MOQ = 60°

Similarly, m∠NOR = 60°

Given, QR =18 cm.

r = 9 cm

**Question 4.Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm.Solution:**

Let S be the length of the arc and r be the radius of the circle.

S = 37.4 cm

Since S = rθ,

**Question 5.A wire of length 10 cm is bent so as to form an arc of a circle of radius 4 cm. What is the angle subtended at the centre in degrees?Solution:**

S = 10 cm and r = 4 cm

Since S = rθ,

10 = 4 x θ

**Question 6.If two arcs of the same length in two circles subtend angles 65° and 110° at the centre. Find the ratio of their radii.Solution:**

Let and be the radii of the two circles and let their arcs of same length S subtend angles of 65° and 110° at their centres.

Angle subtended at the centre of the first circle,

Angle subtended at the centre of the second circle,

**Question 7.The area of a circle is 81TH sq.cm. Find the length of the arc subtending an angle of 300° at the centre and also the area of corresponding sector.Solution:**

**Question 8.Show that minute-hand of a clock gains 5° 30′ on the hour-hand in one minute.Solution:**

Angle made by hour-hand in one minute

[Note: The question has been modified.]

**Question 9.A train is running on a circular track of radius 1 km at the rate of 36 km per hour. Find the angle to the nearest minute, through which it will turn in 30 seconds.Solution:**

r = 1km = 1000m

l(Arc covered by train in 30 seconds)

∴ S = 300 m

Since S = rθ,

300 = 1000 x θ

= (17.18)°

= 17° +(0.18)°

= 17° + (0.18 x 60)’ = 17° + (10.8)’

∴ θ = 17°11′(approx.)

**Question 10.In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.Solution:**

Let ‘O’ be the centre of the circle and AB be the chord of the circle.

Here, d = 40 cm

**Question 11.The angles of a quadrilateral are in A.P. and the greatest angle is double the least. Find angles of the quadrilateral in radians.Solution:**

Let the measures of the angles of the quadrilateral in degrees be a – 3d, a – d, a + d, a + 3d, where a > d > 0

∴ (a – 3d) + (a – d) + (a + d) + (a + 3d) = 360°

… [Sum of the angles of a quadrilateral is 360°]

∴ 4a = 360°

∴ a = 90°

According to the given condition, the greatest angle is double the least,

∴ a + 3d = 2.(a – 3d)

∴ 90° + 3d = 2.(90° – 3d)

∴ 90° + 3d = 180° – 6d 9d = 90°

∴ d = 10°

∴ The measures of the angles in degrees are

a – 3d = 90° – 3(10°) = 90° – 30° = 60°,

a – d = 90° – 10° = 80°,

a + d = 90°+ 10°= 100°,

a + 3d = 90° + 3(10°) = 90° + 30° = 120°