RBSE Solutions Class 9 Maths Chapter 13 Angles and their Measurement More Ques

Multiple Choice Questions (Q1 to Q5)

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: An angle of 750 degrees completes two full circles (720 degrees) and then moves an additional 30 degrees. This extra 30 degrees places the angle in the first quadrant.

🎯 Exam Tip: To find the quadrant of large angles, subtract multiples of 360° until the angle is between 0° and 360°. For negative angles, add multiples of 360°.

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 into radians, we multiply the degree value by \( \frac{\pi}{180^\circ} \). So, 30° becomes \( 30 \times \frac{\pi}{180} = \frac{\pi}{6} \) radians.

🎯 Exam Tip: Remember the conversion factor: \( 1^\circ = \frac{\pi}{180} \) radians and \( 1 \text{ radian} = \frac{180}{\pi} \) degrees. This helps you convert easily.

Question 3. The value of \( \frac {3\pi }{4} \) in sexagesimal system is:
(a) 75°
(b) 135°
(c) 15 minutes
(d) 5 minutes
Answer: (b) 135°
In simple words: To convert radians to degrees (sexagesimal system), you replace \( \pi \) with 180 degrees. So, \( \frac{3 \times 180}{4} \) gives 135 degrees. This method helps to switch between radian and degree measures.

🎯 Exam Tip: When converting from radians to degrees, always substitute \( \pi = 180^\circ \). This simplifies the calculation and reduces errors.

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
(b) 20 minutes
(c) 15 minutes
(d) 5 minutes
Answer: (d) 5 minutes
In simple words: A minute hand moves 360 degrees, or \( 2\pi \) radians, in 60 minutes. So, to find the time for \( \frac{\pi}{6} \) radians, we calculate what fraction of 60 minutes that angle represents. This means \( (\frac{\pi}{6} / 2\pi) \times 60 \) minutes, which simplifies to \( (\frac{1}{12}) \times 60 \) minutes or 5 minutes.

🎯 Exam Tip: Remember that the minute hand covers 360° (or \( 2\pi \) radians) in 60 minutes, which means it moves 6° (or \( \frac{\pi}{30} \) radians) every minute. Use this rate for quick calculations.

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 25 metres is:
(a) \( \frac {\pi }{4} \) radian
(b) \( \frac {\pi }{ 3 } \) radian
(c) \( \frac {\pi }{6} \) radian
(d) \( \frac {3\pi }{ 4 } \) radian
Answer: (a) \( \frac {\pi }{4} \) radian
In simple words: The angle in radians is found by dividing the arc length by the radius. Here, it is \( \frac{25 \text{ metres}}{100 \text{ metres}} = \frac{1}{4} \) radians. Often, \( \frac{1}{4} \) radians is expressed using \( \pi \) for common angle values, like \( \frac{\pi}{4} \approx 0.785 \) radians.

🎯 Exam Tip: The formula for an angle in radians is \( \theta = \frac{\text{arc length (l)}}{\text{radius (r)}} \). Always ensure both length and radius are in the same units for accurate results.

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) 240° is equal to two right angles (180°) plus 60°. This places the ray in the third quadrant.
(ii) 425° is equal to four right angles (360°) plus 65°. This places the ray in the first quadrant.
(iii) -580° is equal to negative six right angles (-540°) minus 40°. This means it's like going 540 degrees clockwise and then an additional 40 degrees clockwise, placing it in the second quadrant.
(iv) 1280° is equal to fourteen right angles (1260°) plus 20°. This places the ray in the third quadrant.
(v) -980° is equal to negative ten right angles (-900°) minus 80°. This means it's like going 900 degrees clockwise and then an additional 80 degrees clockwise, placing it in the second quadrant.
In simple words: To find the quadrant, think of the ray spinning around the origin. A full circle is 360°. If the angle is positive, spin counter-clockwise; if negative, spin clockwise. After completing full circles, see where the remaining angle lands.

🎯 Exam Tip: Remember the quadrant ranges: 1st (0°-90°), 2nd (90°-180°), 3rd (180°-270°), 4th (270°-360°). For angles outside this range, add or subtract multiples of 360° until the angle falls within 0° to 360°.

Question 7. Convert the following angles into radians.
(i) 45°
(ii) 120°
(iii) 135°
(iv) 540°
Answer:
We know that \( 180^\circ = \pi \) radians.
This means \( 1^\circ = \frac{\pi}{180} \) radians.
(i) For 45°: \( 45^\circ = \frac{\pi}{180} \times 45 = \frac{\pi}{4} \) radians.
(ii) For 120°: \( 120^\circ = \frac{\pi}{180} \times 120 = \frac{2\pi}{3} \) radians.
(iii) For 135°: \( 135^\circ = \frac{\pi}{180} \times 135 = \frac{3\pi}{4} \) radians.
(iv) For 540°: \( 540^\circ = \frac{\pi}{180} \times 540 = 3\pi \) radians.
In simple words: To change an angle from degrees to radians, you multiply the degree value by the fraction \( \frac{\pi}{180} \). This fraction is a conversion factor because 180 degrees is the same as \( \pi \) radians.

🎯 Exam Tip: Always simplify the fraction after multiplying by \( \frac{\pi}{180} \) to get the most concise radian measure. Common angles like 30°, 45°, 60°, 90° should be memorized in radians.

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:
We know that \( \pi \) radian \( = 180^\circ \).
(i) For \( \frac{\pi}{2} \): \( \frac{\pi}{2} \) radians \( = \frac{180^\circ}{2} = 90^\circ \).
(ii) For \( \frac{2\pi}{5} \): \( \frac{2\pi}{5} \) radians \( = \frac{2 \times 180^\circ}{5} = 2 \times 36^\circ = 72^\circ \).
(iii) For \( \frac{5\pi}{6} \): \( \frac{5\pi}{6} \) radians \( = \frac{5 \times 180^\circ}{6} = 5 \times 30^\circ = 150^\circ \).
(iv) For \( \frac{\pi}{15} \): \( \frac{\pi}{15} \) radians \( = \frac{180^\circ}{15} = 12^\circ \).
In simple words: To change an angle from radians to the sexagesimal system (degrees), you replace the symbol \( \pi \) with 180 degrees. Then you just do the math to get the angle in degrees.

🎯 Exam Tip: When converting from radians to degrees, simply substitute \( \pi = 180^\circ \) into the expression. This is a direct and efficient method.

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 know that the formula for an angle in radians is:
\( \theta \text{ (radian)} = \frac { \text{arc length} }{ \text{radius} } \)
Given, radius \( r = 5 \) cm and arc length \( l = 12 \) cm.
Substituting these values:
\( \theta = \frac{12}{5} \) radians.
Therefore, the angle subtended is \( \frac{12}{5} \) radians, which is 2.4 radians. This means the angle is a little less than half a full circle.
In simple words: You can find how big an angle is in radians by dividing the length of the arc (the curved part) by the radius (the distance from the center to the edge).

🎯 Exam Tip: Always ensure the arc length and radius are in the same units before applying the formula \( \theta = \frac{l}{r} \). The unit for the angle will be radians.

Question 10. How much time the minute hand of a watch will take to describe an angle of \( \frac {3\pi }{2} \) radians.
Answer:
The minute hand completes a full circle, which is \( 2\pi \) radians, in 60 minutes.
So, in 1 minute, the minute hand covers \( \frac{2\pi}{60} = \frac{\pi}{30} \) radians.
We need to find the time \( t \) (in minutes) for an angle of \( \frac{3\pi}{2} \) radians.
Time \( t = \frac{\text{desired angle}}{\text{rate per minute}} \)
\( t = \frac{3\pi/2}{\pi/30} \)
\( t = \frac{3\pi}{2} \times \frac{30}{\pi} \)
\( t = \frac{3 \times 30}{2} = 3 \times 15 = 45 \) minutes.
So, the minute hand will take 45 minutes to describe an angle of \( \frac{3\pi}{2} \) radians, which is the same as three-quarters of a full turn.
In simple words: The minute hand moves one full circle in 60 minutes. We need to find out what part of a full circle is \( \frac{3\pi}{2} \) radians and then calculate that part of 60 minutes.

🎯 Exam Tip: Remember that a full revolution for a minute hand is 360° or \( 2\pi \) radians, completed in 60 minutes. Use this equivalence to set up a proportion or rate for any angle.

Question 11. How much time the minute hand of a watch will take to describe an angle of 120°?
Answer:
The minute hand of a watch completes a full circle, which is 360°, in 60 minutes.
So, in 1 minute, the minute hand covers \( \frac{360^\circ}{60} = 6^\circ \).
We want to find the time \( t \) (in minutes) for an angle of 120°.
Time \( t = \frac{\text{desired angle}}{\text{rate per minute}} \)
\( t = \frac{120^\circ}{6^\circ/\text{minute}} \)
\( t = 20 \) minutes.
Therefore, the minute hand will take 20 minutes to describe an angle of 120°, which is one-third of a full turn.
In simple words: Since the minute hand moves 6 degrees every minute, to find the time for 120 degrees, you just divide 120 by 6.

🎯 Exam Tip: For problems involving clock hands, always remember the speed of each hand: the minute hand moves 6° per minute, and the hour hand moves 0.5° per minute.

Question 12. Find the radius of the circle, if an arc length of 10 cm subtends an angle of 60° at the centre of the circle.
Answer:
We know the formula relating arc length, radius, and angle:
\( \theta = \frac{\text{arc length}}{\text{radius}} \)
First, we need to convert the angle 60° to radians, because the formula requires the angle in radians.
\( \theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) radians.
Given arc length \( l = 10 \) cm and angle \( \theta = \frac{\pi}{3} \) radians.
We can rearrange the formula to find the radius:
\( \text{radius} = \frac{\text{arc length}}{\theta} \)
\( r = \frac{10}{\pi/3} \)
\( r = \frac{10 \times 3}{\pi} = \frac{30}{\pi} \) cm.
The radius of the circle is \( \frac{30}{\pi} \) cm, which is approximately 9.55 cm. This means the arc is part of a relatively small circle.
In simple words: To find the radius, divide the arc length by the angle, but make sure the angle is in radians first by changing degrees to radians.

🎯 Exam Tip: Always convert angles to radians when using the arc length formula \( l = r\theta \) or \( \theta = l/r \). Failing to do so is a common mistake that leads to incorrect answers.

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 1 right angle is 90°.
So, 30 right angles \( = 30 \times 90^\circ = 2700^\circ \).
The minute hand completes 360° in 60 minutes, which means it rotates 6° per minute.
Time taken \( = \frac{\text{Total angle}}{\text{Rate of minute hand}} = \frac{2700^\circ}{6^\circ/\text{minute}} = 450 \) minutes.
To convert minutes to hours:
\( 450 \text{ minutes} = \frac{450}{60} \text{ hours} = 7.5 \text{ hours} \).
7.5 hours means 7 hours and 30 minutes.
Since the time is just after noon (12:00 p.m.), we add 7 hours and 30 minutes to 12:00 p.m.
12:00 p.m. + 7 hours 30 minutes = 7:30 p.m.
So, the time will be 7:30 p.m. This calculation helps us understand the movement of the clock's minute hand over longer periods.
In simple words: First, figure out the total degrees for 30 right angles. Then, since the minute hand moves 6 degrees each minute, divide the total degrees by 6 to get the minutes. Convert these minutes into hours and add to noon.

🎯 Exam Tip: Convert "right angles" to degrees early in the problem. Then, use the minute hand's speed of 6° per minute to find the time elapsed. Always state whether the final time is A.M. or P.M.

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 \( 4x \).
The sum of angles in a triangle is always 180°.
So, \( 2x + 3x + 4x = 180^\circ \)
\( 9x = 180^\circ \)
\( x = \frac{180^\circ}{9} = 20^\circ \).
Now, we find each angle in degrees:
First angle \( = 2x = 2 \times 20^\circ = 40^\circ \).
Second angle \( = 3x = 3 \times 20^\circ = 60^\circ \).
Third angle \( = 4x = 4 \times 20^\circ = 80^\circ \).
Now, we convert these angles from degrees to radians. We use the conversion factor \( 1^\circ = \frac{\pi}{180} \) radians.
For \( 40^\circ \): \( 40 \times \frac{\pi}{180} = \frac{40\pi}{180} = \frac{2\pi}{9} \) radians.
For \( 60^\circ \): \( 60 \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3} \) radians.
For \( 80^\circ \): \( 80 \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \) radians.
The three angles of the triangle in radians are \( \frac{2\pi}{9} \), \( \frac{\pi}{3} \), and \( \frac{4\pi}{9} \). These radian measures show the relative size of the angles.
In simple words: First, use the ratio to find the angles in degrees, remembering that all angles in a triangle add up to 180 degrees. Then, change each degree angle into radians by multiplying by \( \frac{\pi}{180} \).

🎯 Exam Tip: Always start by finding the value of 'x' using the sum of angles in a triangle (180°) before calculating individual angles. Then, convert each degree measure to radians separately, simplifying fractions for a clean answer.

Question 15. Convert \( \frac {3\pi }{5} \) radian into sexagesimal system.
Answer:
We know that \( \pi \) radian \( = 180^\circ \).
To convert from radians to the sexagesimal system (degrees), we substitute \( \pi \) with \( 180^\circ \).
\( \frac{3\pi}{5} \) radians \( = \frac{3 \times 180^\circ}{5} \)
\( = 3 \times 36^\circ \)
\( = 108^\circ \).
So, \( \frac{3\pi}{5} \) radians is equal to 108 degrees. This conversion helps us to understand the angle in a more familiar unit.
In simple words: To change a radian angle into degrees, simply replace the symbol \( \pi \) with 180 degrees and do the multiplication and division.

🎯 Exam Tip: The conversion \( \pi \text{ radians} = 180^\circ \) is fundamental. Use it directly to convert radian measures to degrees, ensuring you replace \( \pi \) only with its degree equivalent, not its numerical value (like 3.14).