Practical Geometry Solution of CBSE Class 6 Mathematics

Answer: Steps of Construction:
1) Mark a point O on the paper, taking O as center and with a measure of 2.5 cm in the compass, Draw a circle say C.

2) With the same center O and with a measure of 4 cm in the compass, Draw another circle say C.

3) C and C' are required circles with same center and radii 2.5 cm and 4 cm respectively.
Question 3.

Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtaines?

What figure is obtained, if the diameters are perpendicular to each other? How do you check your answer?

Answer:

Step 1: Draw an arbitrary circle using a compass and draw any two diameters.

Step 2: Join the end of diameters and measure the angles with a protractor and lengths with a scale.

Opposite angles are 90 °, and opposide sides are equal. So the polygon formed is a rectangle.

If the diameters are perpendicular, then the adjacent sides are equal, and opposite angles are 90 °s. So the polygon is a square.

Question 4.

Draw any circle and mark points A, B and C such that
(a) A is on the circle.

(b) B is in the interior of the circle.

(c) C is in the exterior of the circle.

Answer:

Step 1: Draw a line segment PQ

Step 2: With P as center draw a circle of radius PQ

Step 3: Extend PQ outwards

P=A is a point on the interior of the circle

Q=B is a point on the circle

C is a point exterior to the circle

Question 5.

Let A, B be the centres of two circles of equal radii; draw them so that each one of them passes through the centre of the other. Let them intersect at C and D.

Examine whether and are at right angles.

Answer:

Step 1: Draw a line segment of any given length

Step 2: With A as center and radius AB draw a circle. With B as a center and radius, BA draw a circle. Join C and D. Measure angle with a protractor.

So AB is perpendicular to CD.

Exercise 14.2

Question 1.

Draw a line segment of length 7.3 cm, using a ruler.

Answer:

Step 1: Draw a line segment of length 7.31cm using a ruler as a reference.

Step 2: Draw a line CD. With A as the centre, take the length of AB with a compass. With C as the centre, cut an arc. Label it as E.

CE=7.3cm

Question 2.

Construct a line segment of length 5.6 cm, using ruler and compasses.

Answer:

Step 1: Draw a line segment of length 5.6cm using a ruler as a reference.

Step 2: Draw a line CD.With A as the centre, take the length of AB with a compass. With C as the centre, cut an arc.Label it as E.

CE=5.6cm

Question 3.

Construct of length 7.8 cm. From this, cut off of length 4.7 cm. Measure .

Answer:

Step 1: Draw line segments of length 7.8cm and 4.7cm using a ruler as a reference.

Step 2: Draw a line AX.With A as the centre, take the length of GD with a compass. With A as the centre, cut an arc.Label it as B.

AB=7.8cm

Step 3: To cut off AC=4.7cm, measure EF with the compass. Cut an arc with A as the centre. Label it as C.

Measure BC.

BC=3.1cm

Question 4.

Given of length 3.9 cm, construct such that the length of is twice that of . Verify by measurement.

Answer:

Steps of Construction-

1. Draw a line segment AB of length 3.9 cm using a ruler.

2. Take a measure of it on the compass and mark a point X.

3. Draw arcs on both sides of X with the same measure in compass and join them named as P and Q.

4. Measure PQ. PQ = 7.8 cm

Question 5.

Given of length 7.3 cm and of length 3.4 cm, construct a line segment such that the length of is equal to the difference between the lengths and . Verify by measurement.

Answer:

Steps of Construction-

1. Draw two lines AB and CD such that AB = 7.3 cm and CD = 3.4 cm

2. Fix the compass pointer on C and and the pencil end on D, the compass now has length of CD

3. Fix the compass pointer on A and swing an arc that cuts AB at E

4. EB is the difference of AB and CD

Verification:
AB - CD = 7.3 - 3.4 = 3.9 cm
On measuring EB = 3.9 cm

Exercise 14.3

Question 1.

Draw any line segment PQ. Without measuring , construct a copy of .

Answer:

Step 1: Draw a line segment PQ

Step 2: With P as centre draw a circle of radius PQ

Step 3: Join P and any point on the circle

PS is thus a copy of PQ.

Question 2.

Given some line segment AB, whose length you do not know, construct such that the length of is twice that of .

Answer:

Step 1: Draw an arbitrary line segment AB

Step 2: With A as centre draw a circle of radius AB

Step 3: Extend AB backwards to meet the circle at Q. BQ=PQ=2AB

Exercise 14.4

Question 1.

Draw any line segment AB. Mark any point M on it. Through M, draw a perpendicular to . (Use ruler and compasses)

s

Answer:

Step 1: Draw an arbitrary line segment AB using a ruler and mark any point M on it

Step 2: With A as center and radius more than AM draw a circle. With B as center and radius the same as before drawing a circle to intersect the first circle at C and D. Join C and D.

Question 2.

Draw any line segment PQ. Take any point R not lying on it. Through R, draw a perpendicular to . (use ruler and set square)

Answer:

Step 1: Draw a line PQ. Point R lies above it.

Step 2: Place a ruler below the line parallel to it. Place the set square on top of the ruler such that its perpendicular edge is at the point R.

Draw a line along the edge, passing through R.

This gives us the required result.

Question 3.

Draw a line l and a point X on it. Through X, draw a line segment XY perpendicular to l.

Now draw a perpendicular to at Y. (use ruler and compasses)

Answer:

Step 1: Draw a line l and mark a point X on it

Step 2: With X as center draw an arc to intersect the line at A and B

Step 3: With A as center and radius more than AX draw an arc. With B as a center and same radius draw an arc to intersect the previous arc at Y. Join XY

XY is perpendicular to AB

Step 4: With Y as center draw an arc intersecting XY at P and Q

Step 5: With P as center and radius more than PY draw an arc. With Q as a center and same radius draw an arc to meet the previous arc at M

Exercise 14.5

Question 1.

Draw of length 7.3 cm and find its axis of symmetry.

Answer:

Step 1: Draw a line segment of length 7.3cm using a ruler

Step 2: With A as center and radius AB draw a circle. With B as a center and same radius draw another circle. Join the point of intersections.

CD is the axis of symmetry

Question 2.

Draw a line segment of length 9.5 cm and construct its perpendicular bisector.

Answer:

Step 1: Draw a line segment 9.5cm using a ruler

Step 2: With A as center and radius AB draw a circle. With B as a center and same radius draw another circle. Join the point of intersections.

CD is the perpendicular bisector.

Question 3.

Draw the perpendicular bisector of whose length is 10.3 cm.

(a) Take any point P on the bisector drawn. Examine whether PX = PY.

(b) If M is the mid-point of , what can you say about the lengths MX and XY?

Answer:

Step 1: Draw line segment XY using a ruler

Step 2: With X as center and radius XY draw a circle. With Y as a center and same radius draw another circle. Join the point of intersections

Step 3: Take any point P on CD. Join PX and PY.Measure PX and PY and show they are equal and is equal to 8cm.

Step 4: Measure MX and XY using a ruler

MX=5.15

XY=10.3

XY=2MX

Question 4.

Draw a line segment of length 12.8 cm. Using compasses, divide it into four equal parts. Verify by actual measurement.

Answer:

Step 1: Draw a line segment of length 12.8cm using a ruler

Step 2: With A as centre and radius AB draw an arc. With B as a centre and same radius draw another arc. Draw a line through their point of intersections.

Step 3: With A as centre and radius AE draw an arc. With E as a centre and same radius draw another arc. Draw a line through their point of intersections

Step 4: With E as centre and radius EB draw an arc. With B as a centre and same radius draw another arc. Draw a line through their point of intersections

AJ, JE,EM,MB are 4 equal parts of AB

Step 5: Measure AJ,JE,EM,MB

Question 5.

With of length 6.1 cm as diameter, draw a circle.

Answer:

Step 1: Draw a line segment of length 6.1 cm using a ruler

Step 2: With P as centre and radius PQ draw a circle. With Q as a centre and same radius draw another circle. Join their point of intersections

Step 3: Through A draw a circle with radius AP

Question 6.

Draw a circle with centre C and radius 3.4 cm. Draw any chord . Construct the perpendicular bisector of and examine if it passes through C.

Answer:

Step 1: Using compass draw a circle of radius 3.4cm and draw any chord AB.

Step 2: With A as centre and radius AB draw a circle. With B as a centre and same radius draw another circle. Join their point of intersection and extend it on either side

Line DE passes through centre C

Question 7.

Repeat question 6, if happens to be a diameter.

Answer:

Step 1: Draw a circle of radius of 3.4 cm and draw any diameter.

Step 2: With A as centre and radius AB draw a circle. With B as a centre and same radius draw another circle. Join their point of intersection and extend it on either side

Thus EF passes through centre C

Question 8.

Draw a circle of radius 4 cm. Draw any two of its chords. Construct the perpendicular bisectors of these chords. Where do they meet?

Answer:

Step 1: Draw a circle of radius 4cm using a compass and draw arbitrary two chords AB and CD

Step 2: With A as centre and radius AB draw An arc. With B as a centre and same radius draw another. Join the point of intersections.

Step 3: With D as centre and radius CD draw a circle. With C as a centre and same radius draw a circle. Join the point of intersections

Step 2 and 3 give the perpendicular bisectors of the chords

So the perpendicular bisectors pass through the centre

Question 9.

Draw any angle with vertex O. Take a point A on one of its arms and B on another such that OA = OB. Draw the perpendicular bisectors of and . Let them meet at P. Is PA = PB.

Answer:

Step 1: Draw a horizontal line and another line inclined arbitrarily to it

Step 2: With O as centre draw a circle to cut OQ at A and OP at B

Step 3: Construct the perpendicular bisectors of OA and OB by constructing circles. Here only the bisector is shown to avoid conjestion.

Step 4: Measure PA and PB using a ruler to show that PA=PB.

Exercise 14.6

Question 1.

Draw ∠POQ of measure 75° and find its line of symmetry.

Answer:

Step 1: Draw an angle of 75 °s using a ruler and protractor

Step 2: With 0 as centre draw an arc to intersect OP and OQ at A and B

Step 3: With centre A and radius momre than AB draw an arc. With B as a centre and same radius draw another arc. Join O with the point of intersection.

Question 2.

Draw an angle of measure 147° and construct its bisector.

Answer:

Step 1: Draw a line AB. Place the centre of the protractor on point A. Coincide AB with the protractor line. Mark C=147.

Step 2: Draw an arc cutting the lines at D and E. Take a radius greater than the arc and from D and E cut to smaller arcs.

Join to form AF being the angle bisector of 147.

Question 3.

Draw a right angle and construct its bisector.

Answer:

Step 1: Draw a line AB. Place the centre of the protractor on point A. Coincide AB with the protractor line. Mark C=90.

Step 2: Take any radius with the compass and draw an arc DE with centre A. From D and E cut 2 arcs with a larger radius.

Join to form AF being the angle bisector of 90.

Question 4.

Draw an angle of measure 153° and divide it into four equal parts.

Answer:

Step 1: Draw a line AB. Place the centre of the protractor on point A. Coincide AB with the protractor line. Mark C=153.

Step 2: Draw an arc cutting AB at D and AC at E.

Taking D as centre and length more than the arc DE cut an arc. Repeat for E.

Step 3: Draw a line through the intersection and extend it to F.

Step 4: Repeat steps 2 and 3 with angle FAB and angle CAF to get the desired answer.

153 is divided into 38.25 each.

Question 5.

Construct with ruler and compasses, angles of following measures:

(a) 60° (b) 30°

(c) 90° (d) 120°

(e) 45° (f) 135°

Answer:

(a) Construction of 60.

Step 1: Draw a line AB using a ruler.

Step 2: Draw an arc of some radius using a compass. Using the same length cut another arc placing the compass on D as centre.

Step 3: Draw a line passing through this point to form AC. Measure angle CAB=60.

(b) Construction of 30.

Construct 60 using the above steps. Bisect the angle as follows:

Step 1: With D as centre draw an arc. With E as centre and same radius draw another arc.

Step 2: Draw a line through their point of intersection.

Step 1: Draw a line AB using a ruler.

Step 2: Draw an arc of some radius using a compass. Using the same length cut another arc placing the compass on D as centre. Repeat with the previous arc.

Step 3: Draw an arc with D and E as centres to get the intersection. Draw a line through their point of intersection.

(d) Construction of 120.

Step 1: Draw a line AB using a ruler.

Step 2: Draw an arc of some radius using a compass. Using the same length cut another arc placing the compass on D as centre. Repeat with the previous arc. Draw a line through the point of intersection to get CA.

(e) Construction of 45.

Follow the steps in 5(c) to get 90. Bisect it to get 45.

(f) Construction of 135.

Step 1: Construct 90.

Step 2: Extend BA to T. With AD as radius draw an arc with centre N and cut AT at G.

Step 3: With G and N as centres draw 2 arcs. Join AM.

Question 6.

Draw an angle of measure 45° and bisect it.

Answer:

Step 1: Draw a line AB. Place the centre of the protractor on point A. Coincide AB with the protractor line. Mark C=45. Join AC.

Step 2: Draw an arc cutting AB at D and AC at E.

Taking D as centre and lenth more than the arc DE cut an arc. Repeat for E.

Step 3: let they intersect at point X. Draw a line joining AX aand extend it to F. Measure angle FAB.

Question 7.

Draw an angle of measure 135° and bisect it.

Answer:

Step 1: Draw a line AB. Place the centre of the protractor on point A. Coincide AB with the protractor line. Mark C=130. Join AC.

Step 2: Draw an arc cutting AB at D and AC at E.

Taking D as centre and lenth more than the arc DE cut an arc. Repeat for E.

Step 3: Let they intersect at point X. Draw a line joining AX aand extend it to F. Measure angle FAB.

Question 8.

Draw an angle of 70°. Make a copy of it using only a straight edge and compasses.

Answer:

Draw a line AB. Place the centre of the protractor on point A. Coincide AB with the protractor line. Mark C=70. Join AC.

Constructing its copy:

Step 1: Draw another line DE.

Step 2: From O, draw an arc cutting at F and G.

Step 3: Draw arc using the compass with the above arc length, placed at D as centre.

Step 4: Measure GF. Using the same compass length place the point at I and cut the arc.

Step 5: using a ruler draw a line through the cut to get DK. Measure angle KDE using a protractor.

Question 9.

Draw an angle of 40°. Copy its supplementary angle.

Answer:

Step 1: Draw any arbitrary line AB and with the help of a compass draw its bisector CD.

Step 2: Measure AE and draw a quarter arc with the compass placing on A.Repeat for EB.

Step 3: With the same measurement draw an arc from E. Join H and I.mark the intersection of CD and HI as J. Join AJ. Measure angle JAB.

Supplementary angle:

Step 1: Extend BA to K. From E, draw an arc of any radius on the supplement of 40.

Step 2: Draw another line OP and using the compass with the above length draw an arc with O as centre.

Step 3: Measure ML. Draw an arc from Q with the measurement of ML. Join to form OT. Measure angle TOP=135.

Practical Geometry

Class 6th Mathematics CBSE Solution

Exercise 14.1

Exercise 14.2

Exercise 14.3

Exercise 14.4

Exercise 14.5

Exercise 14.6

Class 6^th Mathematics CBSE Solution