RBSE Solutions Class 9 Maths Chapter 13 Angles and their Measurement

Multiple Choice Questions

Question 1. The line describing an angle of 750°, lies in:
(a) First quadrant
(b) Second quadrant
(c) Third quadrant
(d) Fourth quadrant
Answer: (a) First quadrant
In simple words: When an angle is 750 degrees, it means the line has gone around the circle more than twice. After going around twice (720 degrees), there are 30 degrees left, which puts it in the first part of the circle.

🎯 Exam Tip: To find an angle's quadrant, subtract 360° repeatedly until the angle is between 0° and 360°, then identify its position.

Question 2. The number of radians in angle 30° is:
(a) \( \frac{\pi}{3} \) radian
(b) \( \frac{\pi}{4} \) radian
(c) \( \frac{\pi}{6} \) radian
(d) \( \frac{3\pi}{4} \) radian
Answer: (c) \( \frac{\pi}{6} \) radian
In simple words: To change degrees to radians, you multiply the angle in degrees by \( \frac{\pi}{180} \). So, 30 degrees becomes \( \frac{30\pi}{180} \), which simplifies to \( \frac{\pi}{6} \) radians. Radians are another way to measure angles.

🎯 Exam Tip: Remember the conversion factor: \( 1^\circ = \frac{\pi}{180} \) radians. Always simplify the fraction to its lowest terms.

Question 3. The value of \( \frac{3\pi}{4} \) in sexagesimal system is:
(a) 75°
(b) 135°
(c) 120°
(d) 220°
Answer: (b) 135°
In simple words: To change radians to degrees, you replace \( \pi \) with 180 degrees. So, \( \frac{3\pi}{4} \) becomes \( \frac{3 \times 180^\circ}{4} \), which works out to 135 degrees. The sexagesimal system uses degrees for angles.

🎯 Exam Tip: When converting radians to degrees, substitute \( \pi \) with 180° directly and perform the calculation. No need to multiply by \( \frac{180}{\pi} \).

Question 4. The time taken by the minute hand of a watch in tracing an angle of \( \frac{\pi}{6} \) radians is:
(a) 10 minutes
(d) None of the options
Answer: (d) None of the options
In simple words: A minute hand moves 360 degrees, or \( 2\pi \) radians, in 60 minutes. To find out how long it takes to move \( \frac{\pi}{6} \) radians, we calculate: \( \frac{\pi/6}{2\pi} \times 60 \) minutes, which is \( \frac{1}{12} \times 60 = 5 \) minutes. Since 5 minutes is not an option, the answer is none of the options.

🎯 Exam Tip: Remember that the minute hand completes a full circle (360° or \( 2\pi \) radians) in 60 minutes, which means it covers \( 6^\circ \) or \( \frac{\pi}{30} \) radians every minute.

Answers

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°
Answer:
(i) For 240°, the ray completes two right angles (180°) and then moves another 60°. This places it in the third quadrant. Quadrants divide the coordinate plane into four sections.
(ii) For 425°, the ray completes four right angles (360°) and then moves another 65°. This places it in the first quadrant.
(iii) For -580°, the ray moves clockwise. It completes six right angles clockwise (-540°) and then moves another 40° clockwise (-540° - 40° = -580°). This places it in the second quadrant.
(iv) For 1280°, the ray completes fourteen right angles (1260°) and then moves another 20°. This places it in the third quadrant.
(v) For -980°, the ray moves clockwise. It completes ten right angles clockwise (-900°) and then moves another 80° clockwise (-900° - 80° = -980°). This places it in the second quadrant.
In simple words: To find the quadrant, you see how many full circles (360 degrees) or half-circles (180 degrees) the ray has turned. Then, look at the remaining angle to find which of the four sections it ends up in. Positive angles turn anti-clockwise, negative angles turn clockwise.

🎯 Exam Tip: Always subtract or add multiples of 360° to bring the angle within 0° to 360° (or -360° to 0° for negative angles) before determining the quadrant. A right angle is 90°.

Question 7. Convert the following angles in radians:
(i) 45°
(ii) 120°
(iii) 135°
(iv) 540°
Answer:
To convert degrees to radians, we multiply the angle by \( \frac{\pi}{180^\circ} \). This ratio helps change the unit of measurement.
(i) For 45°:
\( 45^\circ = 45 \times \frac{\pi}{180^\circ} \text{ radians} \)
\( \implies 45^\circ = \frac{\pi}{4} \text{ radians} \)
(ii) For 120°:
\( 120^\circ = 120 \times \frac{\pi}{180^\circ} \text{ radians} \)
\( \implies 120^\circ = \frac{2\pi}{3} \text{ radians} \)
(iii) For 135°:
\( 135^\circ = 135 \times \frac{\pi}{180^\circ} \text{ radians} \)
\( \implies 135^\circ = \frac{3\pi}{4} \text{ radians} \)
(iv) For 540°:
\( 540^\circ = 540 \times \frac{\pi}{180^\circ} \text{ radians} \)
\( \implies 540^\circ = 3\pi \text{ radians} \)
In simple words: To change an angle from degrees to radians, you always multiply the degree value by \( \pi \) and then divide by 180. Remember that \( \pi \) radians is the same as 180 degrees.

🎯 Exam Tip: Always show the multiplication step with \( \frac{\pi}{180^\circ} \) and simplify the fraction for full marks. Be careful with unit notation.

Question 8. Express the following angles in sexagesimal system.
(i) \( \frac{\pi}{2} \)
(ii) \( \frac{2\pi}{5} \)
(iii) \( \frac{5\pi}{6} \)
(iv) \( \frac{\pi}{15} \)
Answer:
To convert radians to the sexagesimal system (degrees), we multiply the angle by \( \frac{180^\circ}{\pi} \). This is the inverse conversion of radians to degrees.
(i) For \( \frac{\pi}{2} \) radians:
\( \frac{\pi}{2} \text{ radians} = \frac{\pi}{2} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{\pi}{2} \text{ radians} = 90^\circ \)
(ii) For \( \frac{2\pi}{5} \) radians:
\( \frac{2\pi}{5} \text{ radians} = \frac{2\pi}{5} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{2\pi}{5} \text{ radians} = 72^\circ \)
(iii) For \( \frac{5\pi}{6} \) radians:
\( \frac{5\pi}{6} \text{ radians} = \frac{5\pi}{6} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{5\pi}{6} \text{ radians} = 150^\circ \)
(iv) For \( \frac{\pi}{15} \) radians:
\( \frac{\pi}{15} \text{ radians} = \frac{\pi}{15} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{\pi}{15} \text{ radians} = 12^\circ \)
In simple words: When you want to change an angle from radians to degrees, you can just replace \( \pi \) with 180 degrees and then do the math. For example, \( \frac{\pi}{2} \) becomes \( \frac{180}{2} \) degrees.

🎯 Exam Tip: Remember that \( \pi \) in radians represents 180 degrees. This direct substitution simplifies conversions from radians to degrees significantly.

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.
Answer:
We use the formula that connects arc length, radius, and the angle in radians.
The formula is: Angle in radians \( (\theta) = \frac{\text{Arc Length}}{\text{Radius}} \).
Given values are:
Radius \( (r) = 5 \) cm
Arc Length \( (l) = 12 \) cm
Now, we put these values into the formula:
\( \theta = \frac{12}{5} \)
\( \implies \theta = 2.4 \) radians.
The angle at the center is 2.4 radians. This formula is very useful for circular motion problems.
In simple words: To find the angle in radians, just divide the length of the curved part (arc) by the distance from the center to the edge (radius).

🎯 Exam Tip: Ensure that the radius and arc length are in the same units before using the formula. The angle calculated will always be in radians, not degrees.

Question 10. How much time the minute hand of a watch will take to describe an angle of \( \frac{3\pi}{2} \) radians.
Answer:
We know that the minute hand of a watch covers a full circle, which is 4 right angles or \( 2\pi \) radians, in 1 hour.
This means the time taken to trace \( 2\pi \) radians is 1 hour.
So, the time taken to trace 1 radian is \( \frac{1}{2\pi} \) hours.
Now, we need to find the time taken to trace \( \frac{3\pi}{2} \) radians:
Time \( = \left( \frac{1}{2\pi} \right) \times \left( \frac{3\pi}{2} \right) \) hours
\( \implies \text{Time} = \frac{3}{4} \) hours
To convert this to minutes, we multiply by 60:
\( \frac{3}{4} \times 60 = 45 \) minutes.
So, the minute hand takes 45 minutes to describe an angle of \( \frac{3\pi}{2} \) radians. It moves steadily, making this calculation straightforward.
In simple words: The minute hand goes around fully in 60 minutes. We need to find what part of a full turn \( \frac{3\pi}{2} \) radians is, and then find that same part of 60 minutes. \( \frac{3\pi}{2} \) is three-quarters of a full turn, so it takes three-quarters of an hour, which is 45 minutes.

🎯 Exam Tip: Remember that \( 2\pi \) radians is equivalent to 360 degrees or 1 hour of minute hand movement. Convert the given angle to a fraction of the total rotation for easier time calculation.

Question 11. How much time the minute hand of a watch will take to describe an angle of 120°?
Answer:
We know that the minute hand of a watch covers an angle of 360° (or 4 right angles) in 1 hour.
First, we find out what fraction of a full circle 120° is:
Fraction of circle \( = \frac{120^\circ}{360^\circ} = \frac{1}{3} \).
Since a full circle takes 1 hour (60 minutes), \( \frac{1}{3} \) of a full circle will take \( \frac{1}{3} \) of 60 minutes.
Time taken \( = \frac{1}{3} \times 60 \) minutes
\( \implies \text{Time taken} = 20 \) minutes.
So, the minute hand will take 20 minutes to describe an angle of 120°. Understanding the constant speed of clock hands is key here.
In simple words: A minute hand moves all the way around (360 degrees) in 60 minutes. To move 120 degrees, which is one-third of the total circle, it will take one-third of 60 minutes, which is 20 minutes.

🎯 Exam Tip: Convert the angle to a fraction of 360° first, then multiply that fraction by 60 minutes to find the time taken. This method applies to any angle measurement.

Question 12. Find the radius of the circle, if any arc length of 10 cm subtends an angle of 60° at the centre of the circle.
Answer:
We use the formula relating the angle \( \theta \) (in radians), arc length \( l \), and radius \( r \):
\( l = r\theta \)
This means \( r = \frac{l}{\theta} \).
First, we must convert the given angle from degrees to radians:
\( 60^\circ = 60 \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) radians.
Now, we have:
Arc length \( (l) = 10 \) cm
Angle \( (\theta) = \frac{\pi}{3} \) radians
Substitute these values into the formula for radius:
\( r = \frac{10}{\frac{\pi}{3}} \)
\( \implies r = \frac{10 \times 3}{\pi} \)
\( \implies r = \frac{30}{\pi} \) cm.
The radius of the circle is \( \frac{30}{\pi} \) cm. It's crucial to work with radians for this formula.
In simple words: To find the radius, divide the length of the curved part (arc) by the angle at the center, but make sure the angle is in radians first. If the angle is in degrees, change it to radians by multiplying by \( \frac{\pi}{180} \).

🎯 Exam Tip: Always convert angles to radians before using the arc length formula \( l = r\theta \). For calculations, \( \pi \) can be approximated as 3.14 or \( \frac{22}{7} \), but it's often left as \( \pi \) in final answers unless specified.

Question 13. Find the time if the minute hand of a clock has revolved through 30 right angles just after noon.
Answer:
We know that the minute hand of a clock covers 4 right angles (which is 360 degrees or a full circle) in 1 hour.
The minute hand revolved through 30 right angles.
We can express 30 right angles as a multiple of 4 right angles:
\( 30 \text{ right angles} = 7 \times (4 \text{ right angles}) + 2 \text{ right angles} \).
Since 4 right angles take 1 hour:
\( 7 \times (4 \text{ right angles}) \) takes \( 7 \times 1 = 7 \) hours.
And 2 right angles is half of 4 right angles, so it takes \( \frac{1}{2} \) an hour.
So, the total time taken is \( 7 \) hours \( + \frac{1}{2} \) hour \( = 7\frac{1}{2} \) hours.
\( 7\frac{1}{2} \) hours is equal to 7 hours and 30 minutes.
If the hand starts just after noon (12:00 p.m.), then after 7 hours and 30 minutes, the time will be 7:30 p.m. This is a practical application of angles and time.
In simple words: The minute hand spins all the way around four times every hour. If it spins 30 times, we divide 30 by 4 to see how many hours passed, which is 7 and a half hours. Starting at noon, 7 and a half hours later is 7:30 p.m.

🎯 Exam Tip: Clearly define the rotation rate of the minute hand (4 right angles = 1 hour or 360° = 60 minutes) and then convert the total angle into hours and minutes.

Question 14. The angles of a triangle are in the ratio of 2 : 3 : 4. Find all the three angles in radians.
Answer:
Let the angles of the triangle be \( 2x \), \( 3x \), and \( 4x \).
We know that the sum of angles in any triangle is 180°.
So, we can write the equation:
\( 2x + 3x + 4x = 180^\circ \)
\( \implies 9x = 180^\circ \)
\( \implies x = \frac{180^\circ}{9} \)
\( \implies x = 20^\circ \)
Now, we find the measure of each angle in degrees:
First angle \( = 2x = 2 \times 20^\circ = 40^\circ \).
Second angle \( = 3x = 3 \times 20^\circ = 60^\circ \).
Third angle \( = 4x = 4 \times 20^\circ = 80^\circ \).
Next, we convert these angles from degrees to radians. To do this, we multiply each angle by \( \frac{\pi}{180^\circ} \).
For 40°:
\( 40^\circ = 40 \times \frac{\pi}{180^\circ} = \frac{4\pi}{18} = \frac{2\pi}{9} \) radians.
For 60°:
\( 60^\circ = 60 \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) radians.
For 80°:
\( 80^\circ = 80 \times \frac{\pi}{180^\circ} = \frac{4\pi}{9} \) radians.
Thus, the three angles in radians are \( \frac{2\pi}{9} \), \( \frac{\pi}{3} \), and \( \frac{4\pi}{9} \). Understanding angle properties is vital here.
In simple words: First, use the ratio and the fact that a triangle's angles add up to 180 degrees to find each angle in degrees. Then, change each of those degree angles into radians by multiplying them by \( \frac{\pi}{180} \).

🎯 Exam Tip: Always remember that the sum of angles in a triangle is 180°. Clearly show the steps for both finding the angles in degrees and then converting them to radians.

Question 15. Convert \( \frac{3\pi}{5} \) radian into sexagesimal system.
Answer:
To convert an angle from radians to the sexagesimal system (degrees), we multiply it by \( \frac{180^\circ}{\pi} \). This is a standard conversion factor.
Given angle in radians \( = \frac{3\pi}{5} \).
Convert to degrees:
\( \frac{3\pi}{5} \text{ radians} = \frac{3\pi}{5} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{3}{5} \times 180^\circ \)
\( \implies 3 \times 36^\circ \)
\( \implies 108^\circ \).
So, \( \frac{3\pi}{5} \) radians is equal to 108 degrees. This conversion is common in geometry and trigonometry.
In simple words: To change radians to degrees, simply replace the \( \pi \) symbol with 180 degrees and then do the multiplication or division.

🎯 Exam Tip: Remember the basic equivalence: \( \pi \) radians = 180°. Use this to quickly convert between radians and degrees by either substituting \( \pi \) or multiplying by the appropriate conversion factor.