Geometrical Constructions Class-6 RS Aggarwal ICSE Maths Goyal Brothers Prakashan Chapter-22 Solutions. We provide step by step Solutions of Exercise / lesson-22 Geometrical Constructions for ICSE Class-6 RS Aggarwal Mathematics.
Our Solutions contain all type Questions with Exe-22 with Notes on Geometrical Constructions to develop skill and confidence. Visit official Website CISCE for detail information about ICSE Board Class-6 Mathematics.
| Board | ICSE |
| Publications | Goyal brothers Prakshan |
| Subject | Maths |
| Class | 6th |
| Chapter-22 | Geometrical Constructions |
| Writer | RS Aggrawal |
| Book Name | Foundation |
| Topics | Solution of Exe-22 |
| Academic Session | 2021-2022 |
Geometrical Constructions Class-6 RS Aggarwal ICSE Maths Goyal Brothers Prakashan Chapter-22 Solutions
Constructing a 60º Angle
We know that the angles in an equilateral triangle are all 60º in size. This suggests that to construct a 60º angle we need to construct an equilateral triangle as described below.
Step 1: Draw the arm PQ.
Step 2: Place the point of the compass at P and draw an arc that passes through Q.
Step 3: Place the point of the compass at Q and draw an arc that passes through P. Let this arc cut the arc drawn in Step 2 at R.
Step 4 : join P to R the angle QPR is 60
Constructing a 30º Angle
We know that:
30º = 1/2 of 60º
So, to construct an angle of 30º, first construct a 60º angle and then bisect it. Often, we apply the following steps.
Step 1: Draw the arm PQ.
Step 2: Place the point of the compass at P and draw an arc that passes through Q.
Step 3: Place the point of the compass at Q and draw an arc that cuts the arc drawn in Step 2 at R.
Step 4: With the point of the compass still at Q, draw an arc near T as shown.
Step 5: With the point of the compass at R, draw an arc to cut the arc drawn in Step 4 at T.
Step 6: Join T to P. The angle QPT is 30º
Constructing a 120º Angle
We know that: 60 + 120 = 180
This means that 120º is the supplement of 60º. Therefore, to construct a 120° angle, construct a 60° angle and then extend one of its arms
Constructing a 90º Angle
We can construct a 90º angle either by bisecting a straight angle or using the following steps.
Step 1: Draw the arm PA.
Step 2: Place the point of the compass at P and draw an arc that cuts the arm at Q.
Step 3: Place the point of the compass at Q and draw an arc of radius PQ that cuts the arc drawn in Step 2 at R.
Step 4: With the point of the compass at R, draw an arc of radius PQ to cut the arc drawn in Step 2 at S.
Step 5: With the point of the compass still at R, draw another arc of radius PQ near T as shown.
Step 6: With the point of the compass at S, draw an arc of radius PQ to cut the arc drawn in step 5 at T.
Step 7: Join T to P. The angle APT is 90º
Exe-22
Geometrical Constructions Class-6 RS Aggarwal ICSE Maths Goyal Brothers Prakashan Chapter-22 Solutions
Page 260
Question 1:
Construct the following angles using protractor :
(i) 36°
(ii) 58°
(iii) 90°
(iv) 105°
(v) 45°
(vi) 125°
(vii) 20°
(viii) 140°
Answer :
(i) 36°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark point A against 36° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 36°.
(ii) 58°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0 mark of the protractor lying on BC.
(d) On this scale, mark a point A against 58 mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 58°
(iii) 90°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark a point A against 90° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 90°
(iv) 105°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark a point A against 105° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 105°
(v) 45°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark a point A against 45° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 45°
(vi) 125°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark a point A against 125° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 125°
(vii) 20°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark a point A against 20° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 20°
(viii) 140°
Construction :
(a) Draw a ray BC.
(b) Keep the protractor with its central point at B and the horizontal edge along BC.
(c) Look at the scale with 0° mark of the protractor lying on BC.
(d) On this scale, mark a point A against 140° mark and remove the protractor.
(e) Join B and A.
Then, ∠ABC = 140°
Question 2:
Draw ∠AOB = 65° using a protractor Now, using ruler and compasses, construct ∠XOY = ∠AOB.
Answer :
Construction :
(i) Draw any ray OA.
(ii) Construct ∠AOB of measure 65°
(iii) Draw OC, the bisector of ∠AOB. Then, ∠AOC is the desired angle of measure 65°.
Question 3:
Draw ∠XOY= 53° using a protractor. Bisect ∠XOY using ruler and compasses.
Answer :
Construction :
(i) Draw an angle ∠XOY = 53°
(ii) With O as centre draw an arc cutting OX and OB at P and Q respectively.
(iii) With Q on P as centre and radius more than 1/2 of PQ draw arcs which cut each other at L.
Then OL is the bisector of ∠XOY
Question 4:
Draw ∠POQ = 110° using a protector Bisect ∠POQ using ruler and compasses.
Answer:
Construction :
(i) Draw an angle ∠POQ = 110°.
(ii) With O as centre draw an arc cutting OX and OB at P and Q respectively.
(iii) With Q on P as centre and radius more than 1/2 of PQ draw arcs which cut each other at L.
Then OL is the bisector of ∠XOY.
Question 5:
Draw a line segment AB = 5.3 cm. Using ruler and compasses, draw the perpendicular bisector of AB.
Answer :
Construction:
(i) Draw a line AB = 5.3.
(ii) With A and B as centre and radius more than 1/2 AB. Draw two arcs cutting each other at C.
(iii) Join C and D which is the perpendicular bisector of AB.
Question 6:
Draw a line segment AB = 4.8 cm. Take a point P outside AB. From P, draw PQ ⊥ AB, using ruler and compasses.
Answer :
Construction:
(i) With Pas centre and any convenient radius, draw, an arc, cutting AB at M and N.
(ii) With M as centre and radius more than 1/2 MN, draw an arc, below the line AB.
(iii) With N as centre and the same radius, draw another arc, cutting the first arc at Q.
(iv) Join P and Q, meeting AB at X.
Then PQ is the required ray perpendicular to AB.
Question 7:
Draw a line segment AB = 5.2 cm. Take a point C on AB such that AC = 3 Cm. From C, draw CD ⊥ AB using ruler and compasses.
Answer :
Construction :
(i) Draw a line AB = 5.2.
(ii) Take a point AC on AB such that AC = 3 Cm.
(iii) With C as centre draw a semi circle touching line AB at E and F
(iv) With E and F as centre as radius more than 1/2 of E, F. Draw two arc cutting each other at D.
(v) Join CD.
(vi) CD is perpendicular to AB (CD ⊥ AB).
Question 8:
Using ruler and compasses, construct an angle of :
(i) 60°
(ii) 120°
(iii) 30°
(iv) 90°
(v) 45°
Answer :
(i) 60°
Construction:
(a) Draw a line OA of any suitable length.
(b) At 0, draw an arc of any size to cut OA at B.
(c) With B as centre, draw the same size arc C, to cut the previous arc at C.
(d) Join OC and extend up to a suitable point D. Then, ∠DOA 60°
Construction:
(a) With centre O on the line OA, draw an arc to cut this line at C.
(b) With C as centre, drawn a same size arc which cuts the first arc at point D.
(c) With D as centre, draw one more arc of same size which cuts the first arc at E.
(d) Join OE and produce it up to point B. Then, ∠AOB = 120°
(iii) 30°
Construction:
(a) Draw an angle of 60° as drawn as {see in Que. no (i) }
(b) Bisect this angle of get two angles each of 30°. Thus, ∠EOB = 30°
(iv) 90°
Construction :
Let OA be a line and at point O, 90 angle is to be drawn.
(a) With O as centre, draw an arc to cut OA at B.
(b) With B as centre, draw the same size arc to cut the previous arc at C.
(c) Again with C as centre and with the same radius, draw one more arc to cut the first arc at D.
(d)With C and D as centers, draw two arcs of equal radii to cut each other at point E.
(e) Join O and E. Then, ∠AOE = 90°
(v) 45°
Draw an angle of 90 as in question (iv) and bisect it. Each angle so obtained will be 45°.
Question 9:
Draw a line segment AB = 4.6 cm. Construct ∠BAC = 30° and ∠ABC = 60°. Measure ∠ACB
Answer :
∠ACB = 90°
As we know sum of all ∠s of A is 180 in △ABC = ∠CAB + ∠ABC + ∠ACB
180° = 30° + 60° + ∠ACB
= ACB = 180°
∠ACB = 90°
Question 10:
Draw a circle of radius 5 cm using compasses.
Answer :
Construction :
(i) Take any point.
(ii) With the help of compass draw a circle of radius. OP = 5 cm.
Question 11:
Take a point O on a paper. Draw a circle with centre O and radius 4.3 cm.
Answer :
Construction:
(i) Take any point.
(ii) With the help of compass draw a circle of
radius. OP = 4.3 cm.
Question 12:
Mark any two points P and O on a paper Using compasses, draw Circle with centre and passing through Q.
Answer :
Construction :
(i) Take any point two point P and Q.
(ii) With the help of compass draw a circle of radius PO from P: which passes through Q.
Question 13:
Mark any three points O, P, Q. Draw a circle with centre O and which contains P in its interior and Q in its exterior.
Answer :
Construction :
(i) Take three point 0, P, Q.
(ii) From point O draw a circle taking radius between P and Q.
Question 14:
Draw a line segment AB of length 6 cm. Draw a circle with AB as diameter. In this circle draw a chord of length 4 cm. Shade the minor segment of the circle.
Answer :
Construction:
(i) Draw a line segment AB = 6 cm
(ii) Take a point O on AB =1/2 AB = 3 cm.
(iii)Draw a circle from point O whose radius OA = 3 cm.
(iv) With the help of compass from point P cut off PQ = 4 cm. Join the chord PQ.
(v) Now shade minor segment.
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