Construction- 2 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

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Question:8
Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC.
Solution:
We are asked to draw an angle BAC and construct another angle equal to angle BAC
We follow a certain algorithm to construct the angle
Steps of Construction
STEP1: Draw an angle of any measure, and label it as ?BAC. We have to construct another angle equal to this
angle.
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Question:8
Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC.
Solution:
We are asked to draw an angle BAC and construct another angle equal to angle BAC
We follow a certain algorithm to construct the angle
Steps of Construction
STEP1: Draw an angle of any measure, and label it as ?BAC. We have to construct another angle equal to this
angle.
STEP2: Draw a ray LN.
STEP3: With centre A and radius equal to AB, draw an arc cutting the ray AC at point D.
STEP4: With centre L and taking the same radius as AB, draw an arc cutting the ray LN at point P.
STEP5: With centre P, and radius equal to BD, draw an arc intersecting the arc drawn in step 4, at point M.
STEP6: Draw ray LM.
Question:9
Draw an obtuse angle. Bisect it. Measure each of the angles so obtained.
Solution:
We are asked to construct an angle and bisect it and measure each angle
We will follow a certain algorithm to construct
Steps of construction
STEP1: Draw an obtuse angle of any measure. For example take 100 degrees. And label it as ?BAC.
STEP2: With centre as A, and radius of any measure, draw an arc intersecting the ray AB and AC, at L and M
respectively.
STEP3: With centre M and radius greater than half of LM, draw an arc inside the cone.
STEP4: With centre L and taking the same radius as in STEP 3, draw another arc intersecting the previous one at
N.
STEP5: Draw the ray AN. This is the angle bisector of BAC.
After measurement, we can find out that, 
?BAN = ?NAC = 50°.
Question:10
Using your protractor, draw an angle of measure 108°. With this angle as given, draw an angle of 54°.
Solution:
We are asked to draw an angle of 108°, using protractor and obtain an angle of 54°
We will follow certain algorithm for this construction
Steps of construction
STEP1: Using the protractor, draw ?ABC of measure 108°.
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Question:8
Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC.
Solution:
We are asked to draw an angle BAC and construct another angle equal to angle BAC
We follow a certain algorithm to construct the angle
Steps of Construction
STEP1: Draw an angle of any measure, and label it as ?BAC. We have to construct another angle equal to this
angle.
STEP2: Draw a ray LN.
STEP3: With centre A and radius equal to AB, draw an arc cutting the ray AC at point D.
STEP4: With centre L and taking the same radius as AB, draw an arc cutting the ray LN at point P.
STEP5: With centre P, and radius equal to BD, draw an arc intersecting the arc drawn in step 4, at point M.
STEP6: Draw ray LM.
Question:9
Draw an obtuse angle. Bisect it. Measure each of the angles so obtained.
Solution:
We are asked to construct an angle and bisect it and measure each angle
We will follow a certain algorithm to construct
Steps of construction
STEP1: Draw an obtuse angle of any measure. For example take 100 degrees. And label it as ?BAC.
STEP2: With centre as A, and radius of any measure, draw an arc intersecting the ray AB and AC, at L and M
respectively.
STEP3: With centre M and radius greater than half of LM, draw an arc inside the cone.
STEP4: With centre L and taking the same radius as in STEP 3, draw another arc intersecting the previous one at
N.
STEP5: Draw the ray AN. This is the angle bisector of BAC.
After measurement, we can find out that, 
?BAN = ?NAC = 50°.
Question:10
Using your protractor, draw an angle of measure 108°. With this angle as given, draw an angle of 54°.
Solution:
We are asked to draw an angle of 108°, using protractor and obtain an angle of 54°
We will follow certain algorithm for this construction
Steps of construction
STEP1: Using the protractor, draw ?ABC of measure 108°.
STEP2: With centre B, and taking any radius, draw an arc, intersecting the ray BA and the ray BC at point D and E
respectively.
STEP 3: With D and E as centre and radius greater than half of DE, draw arcs to intersect each other at say F.
STEP 4: Draw the ray BF. This is the angle bisector of ?ABC. So ?FBC = 54°.
STEP 5: With centre B and taking any radius, draw an arc intersecting the ray BF and the ray BC at G and H
respectively.
STEP 6: Draw a ray MN.
STEP 7: With centre M and taking the same radius as in STEP 5, draw an arc intersecting the ray MN at point P.
STEP 8: With centre P and taking radius equal to HG, draw an arc intersecting the arc drawn in STEP 7, at point L.
STEP 9: Draw the ray ML.
?LMN is the desired angle of measure 54°.
Question:11
Using protractor, draw a right angle. Bisect it to get an angle of measure 45°.
Solution:
We are asked to construct an angle of 90° using protractor and bisect it
We will follow the following algorithm to construct this figure
Steps of Construction
STEP1: Using the protractor, draw a right angle and label it as ?ABC.
STEP2: With centre as B, draw an arc of any radius, intersecting the ray BA at D and the ray BC at E.
STEP3: With centre as D, and taking radius greater than half of DE, draw an arc inside the ?ABC.
STEP4: With centre as E, and taking the same radius, draw another arc, intersecting the previous arc at F.
STEP5: Draw the ray BF.
Question:12
Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to
each other.
Solution:
We are asked to draw a linear pair of angles and bisect each of them and verify that the two bisecting rays are
perpendicular to each other
We follow the following algorithm for construct
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Question:8
Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC.
Solution:
We are asked to draw an angle BAC and construct another angle equal to angle BAC
We follow a certain algorithm to construct the angle
Steps of Construction
STEP1: Draw an angle of any measure, and label it as ?BAC. We have to construct another angle equal to this
angle.
STEP2: Draw a ray LN.
STEP3: With centre A and radius equal to AB, draw an arc cutting the ray AC at point D.
STEP4: With centre L and taking the same radius as AB, draw an arc cutting the ray LN at point P.
STEP5: With centre P, and radius equal to BD, draw an arc intersecting the arc drawn in step 4, at point M.
STEP6: Draw ray LM.
Question:9
Draw an obtuse angle. Bisect it. Measure each of the angles so obtained.
Solution:
We are asked to construct an angle and bisect it and measure each angle
We will follow a certain algorithm to construct
Steps of construction
STEP1: Draw an obtuse angle of any measure. For example take 100 degrees. And label it as ?BAC.
STEP2: With centre as A, and radius of any measure, draw an arc intersecting the ray AB and AC, at L and M
respectively.
STEP3: With centre M and radius greater than half of LM, draw an arc inside the cone.
STEP4: With centre L and taking the same radius as in STEP 3, draw another arc intersecting the previous one at
N.
STEP5: Draw the ray AN. This is the angle bisector of BAC.
After measurement, we can find out that, 
?BAN = ?NAC = 50°.
Question:10
Using your protractor, draw an angle of measure 108°. With this angle as given, draw an angle of 54°.
Solution:
We are asked to draw an angle of 108°, using protractor and obtain an angle of 54°
We will follow certain algorithm for this construction
Steps of construction
STEP1: Using the protractor, draw ?ABC of measure 108°.
STEP2: With centre B, and taking any radius, draw an arc, intersecting the ray BA and the ray BC at point D and E
respectively.
STEP 3: With D and E as centre and radius greater than half of DE, draw arcs to intersect each other at say F.
STEP 4: Draw the ray BF. This is the angle bisector of ?ABC. So ?FBC = 54°.
STEP 5: With centre B and taking any radius, draw an arc intersecting the ray BF and the ray BC at G and H
respectively.
STEP 6: Draw a ray MN.
STEP 7: With centre M and taking the same radius as in STEP 5, draw an arc intersecting the ray MN at point P.
STEP 8: With centre P and taking radius equal to HG, draw an arc intersecting the arc drawn in STEP 7, at point L.
STEP 9: Draw the ray ML.
?LMN is the desired angle of measure 54°.
Question:11
Using protractor, draw a right angle. Bisect it to get an angle of measure 45°.
Solution:
We are asked to construct an angle of 90° using protractor and bisect it
We will follow the following algorithm to construct this figure
Steps of Construction
STEP1: Using the protractor, draw a right angle and label it as ?ABC.
STEP2: With centre as B, draw an arc of any radius, intersecting the ray BA at D and the ray BC at E.
STEP3: With centre as D, and taking radius greater than half of DE, draw an arc inside the ?ABC.
STEP4: With centre as E, and taking the same radius, draw another arc, intersecting the previous arc at F.
STEP5: Draw the ray BF.
Question:12
Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to
each other.
Solution:
We are asked to draw a linear pair of angles and bisect each of them and verify that the two bisecting rays are
perpendicular to each other
We follow the following algorithm for construct
Steps of construction
STEP1: Draw a linear pair of angles and label them as ABD and DBC.
STEP2: Draw an arc of sufficient radius, intersecting the ray BA, ray BD and the ray BC at points G, H and I
respectively.
STEP3: With centre I and H and radius greater than half of HI draw arcs intersecting each other at point say F.
STEP4: With centre G and H as centres and radius as more than half of GH, draw arcs intersecting each other at
point say E. 
STEP5: Draw the ray BE and the ray BF.
After the measurement, it can be verified that EBF is a right angle.
Hence the ray BE and the ray BF are perpendicular to each other.
Question:13
Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the
same line.
Solution:
We are asked to draw a pair of vertically opposite angles bisect each of them and verify that the two of them are in
the sane line
We will follow the following algorithm for construction
Steps of construction
STEP1: Draw a pair of vertically opposite angles, ABC and DBE.
STEP2: With B as a centre, and taking any radius, draw an arc intersecting the ray BD and the ray BE at points F
and G, respectively.
STEP3: With G as a centre, and radius greater than half of FG, draw an arc inside of DBE.
STEP4: With F as a centre, and taking the same radius as in STEP 3, draw an arc intersecting the arc drawn in
STEP 3, at N.
STE 5: Draw the ray BN.
STEP6: With B as a centre, and taking any radius, draw an arc intersecting the ray BA and the ray BC at points H
and K, respectively.
STEP7: With H as a centre, and radius greater than half of HK, draw an arc inside of ABC.
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Question:8
Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC.
Solution:
We are asked to draw an angle BAC and construct another angle equal to angle BAC
We follow a certain algorithm to construct the angle
Steps of Construction
STEP1: Draw an angle of any measure, and label it as ?BAC. We have to construct another angle equal to this
angle.
STEP2: Draw a ray LN.
STEP3: With centre A and radius equal to AB, draw an arc cutting the ray AC at point D.
STEP4: With centre L and taking the same radius as AB, draw an arc cutting the ray LN at point P.
STEP5: With centre P, and radius equal to BD, draw an arc intersecting the arc drawn in step 4, at point M.
STEP6: Draw ray LM.
Question:9
Draw an obtuse angle. Bisect it. Measure each of the angles so obtained.
Solution:
We are asked to construct an angle and bisect it and measure each angle
We will follow a certain algorithm to construct
Steps of construction
STEP1: Draw an obtuse angle of any measure. For example take 100 degrees. And label it as ?BAC.
STEP2: With centre as A, and radius of any measure, draw an arc intersecting the ray AB and AC, at L and M
respectively.
STEP3: With centre M and radius greater than half of LM, draw an arc inside the cone.
STEP4: With centre L and taking the same radius as in STEP 3, draw another arc intersecting the previous one at
N.
STEP5: Draw the ray AN. This is the angle bisector of BAC.
After measurement, we can find out that, 
?BAN = ?NAC = 50°.
Question:10
Using your protractor, draw an angle of measure 108°. With this angle as given, draw an angle of 54°.
Solution:
We are asked to draw an angle of 108°, using protractor and obtain an angle of 54°
We will follow certain algorithm for this construction
Steps of construction
STEP1: Using the protractor, draw ?ABC of measure 108°.
STEP2: With centre B, and taking any radius, draw an arc, intersecting the ray BA and the ray BC at point D and E
respectively.
STEP 3: With D and E as centre and radius greater than half of DE, draw arcs to intersect each other at say F.
STEP 4: Draw the ray BF. This is the angle bisector of ?ABC. So ?FBC = 54°.
STEP 5: With centre B and taking any radius, draw an arc intersecting the ray BF and the ray BC at G and H
respectively.
STEP 6: Draw a ray MN.
STEP 7: With centre M and taking the same radius as in STEP 5, draw an arc intersecting the ray MN at point P.
STEP 8: With centre P and taking radius equal to HG, draw an arc intersecting the arc drawn in STEP 7, at point L.
STEP 9: Draw the ray ML.
?LMN is the desired angle of measure 54°.
Question:11
Using protractor, draw a right angle. Bisect it to get an angle of measure 45°.
Solution:
We are asked to construct an angle of 90° using protractor and bisect it
We will follow the following algorithm to construct this figure
Steps of Construction
STEP1: Using the protractor, draw a right angle and label it as ?ABC.
STEP2: With centre as B, draw an arc of any radius, intersecting the ray BA at D and the ray BC at E.
STEP3: With centre as D, and taking radius greater than half of DE, draw an arc inside the ?ABC.
STEP4: With centre as E, and taking the same radius, draw another arc, intersecting the previous arc at F.
STEP5: Draw the ray BF.
Question:12
Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to
each other.
Solution:
We are asked to draw a linear pair of angles and bisect each of them and verify that the two bisecting rays are
perpendicular to each other
We follow the following algorithm for construct
Steps of construction
STEP1: Draw a linear pair of angles and label them as ABD and DBC.
STEP2: Draw an arc of sufficient radius, intersecting the ray BA, ray BD and the ray BC at points G, H and I
respectively.
STEP3: With centre I and H and radius greater than half of HI draw arcs intersecting each other at point say F.
STEP4: With centre G and H as centres and radius as more than half of GH, draw arcs intersecting each other at
point say E. 
STEP5: Draw the ray BE and the ray BF.
After the measurement, it can be verified that EBF is a right angle.
Hence the ray BE and the ray BF are perpendicular to each other.
Question:13
Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the
same line.
Solution:
We are asked to draw a pair of vertically opposite angles bisect each of them and verify that the two of them are in
the sane line
We will follow the following algorithm for construction
Steps of construction
STEP1: Draw a pair of vertically opposite angles, ABC and DBE.
STEP2: With B as a centre, and taking any radius, draw an arc intersecting the ray BD and the ray BE at points F
and G, respectively.
STEP3: With G as a centre, and radius greater than half of FG, draw an arc inside of DBE.
STEP4: With F as a centre, and taking the same radius as in STEP 3, draw an arc intersecting the arc drawn in
STEP 3, at N.
STE 5: Draw the ray BN.
STEP6: With B as a centre, and taking any radius, draw an arc intersecting the ray BA and the ray BC at points H
and K, respectively.
STEP7: With H as a centre, and radius greater than half of HK, draw an arc inside of ABC.
STEP8: With K as a centre, and taking the same radius as in STEP 7, draw an arc intersecting the arc drawn in
STEP 7, at M
STEP9: Draw the ray BM
After measuring the MBN, it can be verified that the bisecting rays BM and BN are in the same line.
Question:14
Using ruler and compasses only, draw a right angle.
Solution:
We have to draw a right angle using ruler and compasses only
We use the following algorithm for the construction
Steps of construction
STEP1: Draw a ray BC.
STEP2: With B as a centre, and taking convenient radius, draw an arc, intersecting the ray BC at point N.
STEP3: With N as a centre, and taking the same radius, draw an arc cutting the previous arc at M.
STEP4: With M as a centre, and the same radius, draw an arc cutting the arc drawn in STEP2 at L.
STEP5: With M and L as centre and the same radius draw arcs intersecting at a point say A.
STEP6: Draw the ray BA.
ABC is a right angle.
Question:15
Using ruler and compasses only, draw an angle of measure 135°.
Solution:
We have to draw an angle of 135° using ruler and compasses only
We follow the following algorithm for the construction
Steps of construction
The below given steps will be followed to construct an angle of 135°.