RBSE Maths Chapter 13 – Angles and their Measurement Class 9 Important questions and solutions are available here. The important questions and solutions of Chapter 13, provided at BYJU’S, have detailed steps. Also, RBSE Class 9 solutions are given at BYJU’S to help the students in getting maximum marks in their exams and in improving their problem-solving skills.
Chapter 13 of the RBSE Class 9 Maths has formulas and various problems on the conversion of angles from degrees to radians, radians to degrees, finding the length of the arc or radius of the circle or angle subtended at the centre of the circle. Also, students will be able to understand the time taken by the minute hand of a clock in describing a certain measure of the angle.
RBSE Maths Chapter 13: Exercise 13 Textbook Important Questions and Solutions
Question 1: The line describing an angle of 750° lies in:
(A) First quadrant
(B) Second quadrant
(C) Third quadrant
(D) Fourth quadrant
Solution:
Correct answer: (A)
750° = 720° + 30°
= 2(360°) + 30°
360° means one complete angle and 30° lies in the first quadrant.
Therefore, 750° lies in the first quadrant.
Or
750° = 720° + 30° = 8 x right angles + 30°
Therefore, the position of the revolving ray of a given angle will be in the first quadrant.
Question 2: The number of radians in angle 30° is:
(A) π/3 radian
(B) π/4 radian
(C) π/6 radian
(D) 3π/4 radian
Solution:
Correct answer: (C)
Given angle is 30°
We know that 1° = π/180° radians
30° = (π/180°) × 30°
= π/6
Therefore, 30° = π/6 radians
Question 3: The value of 3π/4 in sexagesimal system is:
(A) 75°
(B) 135°
(C) 120°
(D) 220°
Solution:
Correct answer: (B)
Given,
Measure of angle in radians = 3π/4
Using sexagesimal system,
D = (180°/π) × R
= (180°/π)× (3π/4)
= (180° × 3)/4
= 135°
Therefore, the degree value of 3π/4 is 135°.
Question 4: How much time will the minute hand of a watch take to describe an angle of π/6 radians?
(A) 10 minutes
(B) 20 minutes
(C) 15 minutes
(D) 5 minutes
Solution:
Correct answer: (D)
Given, the measure of angle is π/6.
D = (180°/π) × R
= (180°/π)× (π/6)
= 180°/6
= 30°
Therefore, the degree of π/6 is 30°.
We know that the minute hand of a clock describes 6° in one minute.
Hence, time taken by the minute hand to describe 30° = 30°/6° = 5 minutes
Question 5: The value of the angle, in radian, subtended at the centre of the circle of radius 100 metres by an arc of length 25π metres is:
(A) π/4
(B) π/3
(C) π/6
(D) 3π/4
Solution:
Correct answer: (A)
Given,
Radius of the circle = r = 100 metres
Length of the arc = l = 25π metres
Angle = l/r = 25π/100 = π/4
Question 6: In which quadrant does the revolving ray lie when it makes the following angles?
(i) 240° (ii) 425° (iii) -580° (iv) 1280° (v) -980°
Solution:
(i) 240° = 180° + 60° = 2 x right angle + 60°
Therefore, the position of the revolving ray will be in the third quadrant.
(ii) 425° = 360° + 65° = 4 x right angles + 65°
Therefore, the position of the revolving ray will be in the first quadrant.
(iii) – 580° = – 540° + (-40°) = – 6 x right angles – 40°
Therefore, the position of the revolving ray will be in the second
quadrant.
(iv) 1280° = 1260° + 20° = 14 x right angles + 20°
Therefore, the position of the revolving ray will be in the third quadrant.
(v) – 980° = -900° – 80° = – 10 x right angles – 80°
Therefore, the position of the revolving ray will be in the second quadrant.
Question 7: Convert the following angles in radians:
(i) 45° (ii) 120° (iii) 135° (iv) 540°
Solution:
We know that 1° = π/180° radians
(i) 45° = (π/180°) × 45° = π/4 radians
(ii) 120° = (π/180°) × 120° = 2π/3 radians
(iii) 135° = (π/180°) × 135° = 3π/4 radians
(iv) 540° = (π/180°) × 540° = 3π radians
Question 8: Express the following angles in sexagesimal system:
(i) π/2 (ii) 2π/5 (iii) 5π/6 (iv) π/15
Solution:
Using sexagesimal system,
D = (180°/π) × R
D = Measure of angle in degrees
R = Measure of angle in radians
(i) π/2 = (180°/π) × (π/2) = 90°
(ii) 2π/5 = (180°/π) × (2π/5) = 72°
(iii) 5π/6 = (180°/π) × (5π/6) = 150°
(iv) π/15 = (180°/π) × (π/15) = 12°
Question 9: Find the angle in radians, subtended at the centre of a circle of radius 5 cm by an arc of the circle whose length is 12 cm.
Solution:
Given,
Radius of the circle = r = 5 cm
Length of the arc = l = 12 cm
Angle = l/r = 12/5 radians
Question 10: How much time will the minute hand of a watch take to describe an angle of 3π/2 radians?
Solution:
Given, the measure of angle is 3π/2 radians.
We know that,
D = (180°/π) × R
= (180°/π) × (3π/2)
= 270°
Therefore, the degree value of 3π/2 is 270°.
We know that the minute hand of a clock describes 6° in 1 minute.
Hence, time taken by the minute hand to describe 270° = 270°/6° = 45 minutes
Question 11: How much time will the minute hand of a watch take to describe an angle of 120°?
Solution:
Minute hand of a clock describes 360° in 1 hour (i.e. 60 minutes).
Angle describes in 1 minute = 360°/60 = 6°
Therefore, time taken by the minute hand to describe 120° = 120°/6° = 20 minutes
Question 12: In a circle, the angle subtended at the centre by an arc of length 10 cm is 60°. Find the radius of the circle.
Solution:
Given,
Length of an arc = l = 10 cm
Let r be the radius of the circle.
Angle = 60°
Measure of angle in radians = (π/180) × 60 = π/3
We know that,
Measure of angles (in radians) = l/r
π/3 = 10/r
r = 10 × (3/π) = 30/π
Therefore, Radius of the circle = 30/π cm
Question 13: If the minute hand of a watch has revolved through 30 right angles, just after the mid day, then what is the time by the watch?
Solution:
We know that the time taken by the minute hand of a clock in revolving 4 right angles is 1 hour (i.e. 60 minutes).
Converting 30 right angles in terms of the multiple of 4 right angles as shown below:
30 right angles = 7 x (4 right angles) + 2 right angles
= 7 x 1 hr + 1/2 hr
= 7 hours 30 minutes
Since the time is considered after mid day i.e. after 12 pm.
Therefore, time shown on the watch is 7:30 pm.
Question 14: The angles of a triangle are in ratio 2 : 3 : 4. Find all three angles in radians.
Solution:
Given ratio of three angles of a triangle is 2 : 3 : 4.
Let 2x, 3x and 4x be the angles of the triangle.
By the angle sum property of a triangle,
2x + 3x + 4x = 180°
9x = 180°
x = 180°/9 = 20°
Thus, angles of the triangle are:
2x = 2(20°) = 40°
3x = 3(20°) = 60°
4x = 4(20°) = 80°
We know that 1° = π/180° radians
Therefore, angles of a triangle in radians are:
40° = (π/180°) × 40° = 2π/9 radians
60° = (π/180°) × 60° = π/3 radians
80° = (π/180°) × 80° = 4π/9 radians
Question 15: Express 3π/5 into a sexagesimal system.
Solution:
Given,
Measure of angle in radians = 3π/5
Using sexagesimal system,
D = (180°/π) × R
= (180°/π)× (3π/5)
= (180° × 3)/5
= 108°
Therefore, the degree value of 3π/5 is 108°.
RBSE Maths Chapter 13: Additional Important Questions and Solutions
Question 1: Change π/4 radian into degrees.
Solution:
Measure of angle in radians = π/4
D = Measure of angle in degrees
Using sexagesimal system,
D = (180°/π) × R
= (180°/π) × (π/4)
= 45°
Therefore, the degree value of π/4 is 45°.
Question 2: Convert 60° in radians.
Solution:
Given angle is 60°
We know that 1° = π/180° radians
60° = (π/180°) × 60°
= π/3
Therefore, 60° = π/3 radians
Question 3: Convert 100° in radians.
Solution:
Given angle is 100°
We know that 1° = π/180° radians
100° = (π/180°) × 100°
= 5π/9
Therefore, 100° = 5π/9 radians
Question 4: Express 5π/2 in degrees.
Solution:
Measure of angle in radians = 5π/2
D = Measure of angle in degrees
Using sexagesimal system,
D = (180°/π) × R
= (180°/π) × (5π/2)
= (180° × 5)/2
= 450°
Therefore, the degree value of 5π/2 is 450°.
Question 5: Find the length of the arc subtending an angle of π/3 radians at the centre of the circle of radius 60 cm.
Solution:
Given,
Radius of the circle = r = 60 cm
Let l be the length of the arc.
Angle subtended at the centre of the circle = π/3 radians
Measure of angle in radians = l/r
π/3 =l/60
l = 60 × (π/3)
l = 20π cm
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