RBSE Solutions Class 9 Maths Chapter 5 Plane Geometry and Line and Angle Exercise 5.3

Chapter 5 Plane Geometry and Line and Angle Ex 5.3

Question 1. Draw a line segment AB = 10 cm. Bisect this line segment and verify by measuring the lengths of the two segments.
Answer:
**Steps of Construction:**
1. First, draw a line segment AB that is 10 cm long.
2. Next, place the compass point on A and open it to a radius that is a bit more than half of AB. Draw arcs above and below the line segment AB.
3. Without changing the compass radius, place the compass point on B and draw more arcs above and below AB. Make sure these new arcs cross the first set of arcs. Label the crossing points M and N.
4. Draw a straight line connecting M and N. This line will cut the line segment AB at a point, let's call it L.
5. When you measure the lengths, you will find that AL = BL = 5 cm, meaning the line segment AB has been divided into two equal parts.
In simple words: First, draw a 10 cm line. Then, use a compass to draw crossing arcs from both ends. Draw a line through where the arcs cross, and it will cut the first line exactly in half. Each half will be 5 cm.

🎯 Exam Tip: Remember to use a compass opening slightly more than half the line segment length to ensure the arcs intersect clearly.

Question 2. Construct an angle of 120°. Bisect the angle and verify by measuring the two angles.
Answer:
**Steps of Construction:**
1. Draw a line segment PQ of any chosen length.
2. Place the protractor's center on point P and mark a 120° angle. Draw a ray from P through this mark. Let's call this ray PS, so that ∠SPQ = 120°.
3. Now, to bisect the angle, place the compass point on P and draw an arc that cuts both ray PQ (at point M) and ray PS (at point N). This creates a section of the angle to work with.
4. Next, place the compass point on M and draw an arc within the angle. Without changing the radius, place the compass point on N and draw another arc that crosses the first one. Label the intersection point T.
5. Draw a ray from P through point T. This ray PT is the bisector of ∠SPQ.
6. If you measure, you will find that ∠TPQ = ∠TPS = 60° each, which confirms the angle is bisected.
In simple words: First, draw an angle of 120 degrees. Then, use a compass to draw an arc inside the angle from its vertex. From the two points where this arc crosses the angle's arms, draw two more arcs that cross each other. Drawing a line from the vertex to where these arcs meet will cut the 120-degree angle into two 60-degree angles.

🎯 Exam Tip: For angle bisection, always make sure your compass arcs from M and N have a radius slightly more than half the MN distance to ensure clear intersection.

Question 3. Draw an angle of 40° using protractor. Now, with the help of ruler and compass, construct an angle equal to this angle.
Answer:
**Steps of Construction:**
1. Draw a ray AB. Place your protractor on point A and mark 40°. Draw another ray AC through this mark, creating ∠BAC = 40°.
2. With A as the center and any convenient radius, draw an arc that cuts ray AB at point P and ray AC at point Q.
3. Now, draw a new line segment EF. Pick a point O on this line segment to be the vertex of your new angle.
4. With O as the center and using the *same radius* as the arc PQ from step 2, draw a new arc that cuts the line segment OF at point R.
5. Open your compass to the length of the arc PQ (from point P to point Q on the first angle). With R as the center and this new radius (PQ), draw an arc that crosses the arc you drew in step 4. Label this new intersection point S.
6. Draw a ray from O through point S. The angle ∠ROS is now equal to 40°, copied from ∠BAC.
In simple words: First, draw a 40-degree angle and mark an arc. Then, on a new line, draw an arc with the same opening. Measure the distance across the arc of your first angle with a compass and use that measure to make a new arc on the second line. Connect the vertex to this new crossing point, and you will have copied the 40-degree angle.

🎯 Exam Tip: The key to constructing an equal angle is ensuring that the radius of the initial arc and the measurement of the chord (distance between P and Q) are exactly the same for both angles.

Question 4. Draw a line segment of length 6 cm. From a point P outside this line, draw a perpendicular on this line.
Answer:
**Steps of construction:**
1. First, draw a line segment AB that is 6 cm long.
2. Choose a point P that is outside this line segment. From point P, draw an arc that cuts the line AB at two distinct points. Let's call these points C and D.
3. Now, using C as the center and a suitable radius (more than half of CD), draw an arc below line AB. Then, with D as the center and the same radius, draw another arc that crosses the first one. Label their intersection point Q.
4. Draw a line segment connecting P and Q. This line segment PQ will be perpendicular to AB from point P.
In simple words: Draw a 6 cm line. Pick a point outside it. From that point, draw an arc that cuts the line in two places. From those two places, draw two more arcs below the line that cross. Connect the outside point to where these two arcs cross, and that line will be straight up-and-down (perpendicular) to the first line.

🎯 Exam Tip: When drawing arcs from C and D, make sure the radius is wide enough (more than half of CD) for them to intersect clearly, which is crucial for accuracy.

Question 5. Construct ∠ABC = 120°. Through A draw a line parallel to BC.
Answer:
**Steps of construction:**
1. First, draw a line segment BC of any chosen length.
2. Place your compass on point B and draw an angle of 120° using a compass and ruler (by marking 60° twice from BC). Let the ray forming this angle be BA, so ∠ABC = 120°.
3. Now, consider point A as a separate point, which lies on the arm of the 120° angle. To draw a line parallel to BC through A, we will construct an angle of 60° at point A (such that it forms a co-interior angle with the 120° at B, making their sum 180°). This new ray from A will be parallel to BC.
In simple words: Draw a line segment BC. At point B, make a 120-degree angle. Now, from point A (which is on the other arm of the angle at B), draw a 60-degree angle so that it looks like the angle is "inside" the space between the two lines. The line you draw from A will be parallel to BC.

🎯 Exam Tip: When constructing parallel lines using transversals, remember that co-interior angles must add up to 180°, which means if one is 120°, the other must be 60°.

Question 6. Draw a line segment of length 9 cm using ruler and compass. Divide this line segment into three equal parts.
Answer:
**Steps of construction:**
1. Draw a line segment AB that is 9 cm long.
2. At point A, draw a ray AX that forms an acute angle (less than 90°) with AB.
3. Using a compass, mark three equal distances along the ray AX, starting from A. Label these points \( A_1, A_2, A_3 \). Each segment ( \( AA_1 \), \( A_1A_2 \), \( A_2A_3 \) ) should be of the same length.
4. Draw a line segment connecting \( A_3 \) to point B.
5. Now, draw lines through \( A_1 \) and \( A_2 \) that are parallel to \( A_3B \). To do this, construct angles equal to \( \angle AA_3B \) at \( A_1 \) and \( A_2 \). These parallel lines will intersect AB at points C and D.
6. You will find that AC = CD = DB = 3 cm each. This means AB is divided into three equal parts.
In simple words: Draw a 9 cm line. From one end, draw a sloping line. Mark three equal steps along this sloping line. Connect the third step to the other end of your main line. Then, draw lines from the first and second steps that are parallel to this connecting line. These parallel lines will divide your 9 cm line into three equal parts, each 3 cm long.

🎯 Exam Tip: Ensure the equal distances marked on the ray are consistent, as this directly impacts the accuracy of the division of the main line segment.

Question 7. Draw a line segment of length 10 cm. Using ruler and compass divide this segment in the ratio 3: 2. Measure the length of these segments.
Answer:
**Steps of construction:**
1. Draw a line segment AB that is 10 cm long.
2. At point A, draw a ray AY that forms an acute angle (less than 90°) with AB.
3. Since the ratio is 3:2, you need a total of \( 3 + 2 = 5 \) equal parts. Using your compass, mark five equal distances along ray AY, starting from A. Label these points P, Q, R, S, T. (So, AP=PQ=QR=RS=ST).
4. Draw a line segment connecting the last point, T, to point B.
5. Now, you need to divide in the ratio 3:2, so you'll draw a parallel line from the 3rd point, R. Through point R, draw a line parallel to TB. This parallel line (let's call it RL) will intersect AB at point L.
6. When you measure, you will find that AL = 6 cm and LB = 4 cm, which correctly divides AB in the ratio 3:2.
In simple words: Draw a 10 cm line. From one end, draw a sloping line and mark 5 equal points on it. Connect the fifth point to the other end of the 10 cm line. Then, from the third point on your sloping line, draw a line parallel to the first diagonal line. This parallel line will cut your 10 cm line into two parts: one 6 cm long and the other 4 cm long.

🎯 Exam Tip: The number of equal parts to mark on the ray is the sum of the ratio parts (e.g., 3:2 means 5 parts), and the parallel line is drawn from the point corresponding to the first part of the ratio (e.g., the 3rd point for 3:2).

Question 8. Draw a line segment AB of length 6 cm, using ruler and compass divide this in the ratio 1:2:3.
Answer:
**Steps of construction:**
1. Draw a line segment AB that is 6 cm long.
2. At point A, draw a ray AX that forms an acute angle (less than 90°) with AB.
3. The ratio is 1:2:3, so you need a total of \( 1 + 2 + 3 = 6 \) equal parts. Using your compass, mark six equal distances along ray AX, starting from A. Label these points \( A_1, A_2, A_3, A_4, A_5, A_6 \).
4. Draw a line segment connecting the last point, \( A_6 \), to point B.
5. Now, you need to divide the line in the ratio 1:2:3. This means you will need parallel lines from \( A_1 \) and \( A_3 \). Draw a line through \( A_1 \) parallel to \( A_6B \), intersecting AB at N. Then draw another line through \( A_3 \) parallel to \( A_6B \), intersecting AB at M.
6. You will find that AN:NM:MB = 1:2:3, dividing the 6 cm line into 1 cm, 2 cm, and 3 cm segments respectively.
In simple words: Draw a 6 cm line. From one end, draw a sloping line and mark 6 equal points on it. Connect the sixth point to the other end of the 6 cm line. Now, draw parallel lines from the first and third marked points on the sloping line. These parallel lines will divide your 6 cm line into parts that match the 1:2:3 ratio (1 cm, 2 cm, and 3 cm).

🎯 Exam Tip: For dividing a line in a ratio like 1:2:3, remember that the total number of equal parts on the auxiliary ray is the sum of the ratio numbers. The parallel lines are then drawn from the corresponding points to create the correct segments.

Question 9. Using ruler and compass, construct the following angles.
(i) 150°
(ii) 105°
(iii) 75°
Answer:
(i) **Construction of 150°:**
**Steps of Construction:**
1. Draw a straight line segment AB. Let O be any point on this line.
2. With O as the center and any convenient radius, draw a semicircle that cuts OA at X and OB at Y. This forms a 180° straight angle.
3. With Y as the center and the same radius, draw an arc that cuts the semicircle at Z. The angle ∠YOZ is 60°.
4. With X as the center and the same radius, draw an arc that cuts the semicircle at M. The angle ∠XOM is 60°.
5. To construct 150°, we can use the concept of 180° - 30°. Bisect the angle ∠XOM (60°) to obtain 30°. Let the bisector be ray OP.
6. The angle ∠POY will then be \( 180° - 30° = 150° \).

(ii) **Construction of 105°:**
**Steps of Construction:**
1. Draw a line segment OB. With O as the center and any radius, draw an arc that cuts OB at point X.
2. From X, with the same radius, cut the arc at Y (creating 60°). From Y, with the same radius, cut the arc at Z (creating 120°).
3. Bisect the arc YZ to construct 90°. Let the ray for 90° be OX' (this is a dashed line in the diagram).
4. Now, bisect the angle between the 90° ray (OX') and the 120° mark (Z from the initial arc). This bisector will add 15° to the 90° angle.
5. Draw a ray OP that forms \( 90° + 15° = 105° \) with OB. The angle ∠BOP is the required 105° angle.

(iii) **Construction of 75°:**
**Steps of Construction:**
1. Draw a line segment OB. With O as the center and any radius, draw an arc that cuts OB at point X.
2. From X, with the same radius, cut the arc at Y (creating 60°). From Y, with the same radius, cut the arc at Z (creating 120°).
3. Bisect the arc YZ to construct 90°. Let the ray for 90° be OX'.
4. Now, bisect the angle between the 60° mark (Y from the initial arc) and the 90° ray (OX'). This bisector will add 15° to the 60° angle.
5. Draw a ray OP that forms \( 60° + 15° = 75° \) with OB. The angle ∠BOP is the required 75° angle.

In simple words: To make these angles, start by drawing a line. Then, use a compass to mark 60-degree points on an arc. For 150 degrees, make a 30-degree angle from the 180-degree line. For 105 degrees, make a 90-degree angle and then add another 15 degrees to it. For 75 degrees, make a 60-degree angle and then add another 15 degrees.

🎯 Exam Tip: Mastering the construction of 60°, 90°, and 120° angles is fundamental, as most other common angles (like 15°, 30°, 45°, 75°, 105°, 150°) are derived by bisecting or combining these basic angles.