Q 1.

Draw a circle of radius 3.2 cm.

SOLUTION:

Steps of construction :
(i) Mark a point 'O' with a sharp pencil where we want the centre of the circle.
(ii) Open the compasses for the required radius of 3.2 cm.
(iii) Place the pointer of compasses on O.
(iv) Turn the compass slowly to draw the circle.

Q 2.

With the same centre O, draw two circles of radii 4 cm and 2.5 cm.

SOLUTION:

Steps of construction :
(i) Mark a point 'O' with a sharp pencil where we want the centre of the circle.
(ii) Open the compass for 4 cm.
(iii) Place the pointer of the compass on O.
(iv) Turn the compass slowly to draw the circle.
(v) Again open the compass for 2.5 cm and place the pointer of the compass on O.
(vi) Turn the compass slowly to draw the second circle.

Q 3.

Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? What figure is obtained if the diameters are perpendicular to each other? How do you check your answer?

SOLUTION:

Steps of construction:
(i) Mark a point 'O' with sharp pencil where we want the centre of the circle.
(ii) Place the pointer of the compasses on 'O'.
(iii) Turn the compasses slowly to draw the circle.
(A) By joining the ends of two diameters, we get a rectangle. By measuring, we findAB = CD and BC = AD, i.e., pairs of opposite sides are equal and also ∠A = ∠B = ∠C = ∠D = 90°, i.e., each angle is of 90°.
Hence, it is a rectangle.

(B) If the diameters are perpendicular to each other, then by joining the ends of two diameters, we get a square.
By measuring, we find that AB = BC = CD = DA, i.e., all four sides are equal.
Also ∠A = ∠B = ∠C = ∠D = 90°, i.e., each angle is 90°.
Hence, it is a square.

Q 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.

SOLUTION:

Steps of construction:
(i) Mark a point 'O' with sharp pencil where we want centre of the circle.
(ii) Place the pointer of the compasses on 'O'.
(iii) Turn the compasses slowly to draw a circle.
(A) Point A is on the circle.
(B) Point B is in the interior of the circle.
(C) Point C is in the exterior of the circle.

Q 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.

SOLUTION:

Steps of construction :
(i) Mark a point 'A' with sharp pencil.
(ii) Place the pointer of the compasses on A.
(iii) Turn the compasses slowly to draw a circle with centre A.
(iv) Take a point 'B' on the circle with centre A.
(v) Place the pointer of the compasses on B.
(vi) Draw the circle with the radius same as radius of circle with centre A.

Now, both circles intersect at C and D.
Let AB and CD intersect at O.
Now, measuring ∠COB, we get
∠COB = 90°
∴ and are perpendicular.

Q 6.

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

SOLUTION:

Steps of construction :

(i) Place the zero mark of the ruler at a point A.
(ii) Mark a point B at a distance of 7.3 cm from A.
(iii) Join AB.
is the required line segment of length7.3 cm.

Q 7.

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

SOLUTION:

Steps of construction :

(i) Draw a line l. Mark a point A on this line.
(ii) Place the compasses pointer on zero mark of the ruler. Open it to place the pencil point upto 5.6 cm mark.
(iii) Without changing the opening of the compasses, place the pointer on A and draw an arc that cuts l at B.
is the required line segment of length 5.6 cm.

Q 8.

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

SOLUTION:

Steps of construction :

(i) Place the zero mark of the ruler at A.
(ii) Mark a point B at a distance 7.8 cm from A.
(iii) Again, mark a point C at a distance 4.7 from A.
(iv) By measuring , we find that BC = 3.1 cm.

Q 9.

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

(Hint: Construct such that length of = length of ; then cut off such that also has the length of .)

SOLUTION:

Steps of construction :

(i) Draw a line 'l'.
(ii) Construct such that length of = lenght of = 3.9 cm
(iii) Then cut of such that also has the length of .
(iv) Thus the length of and the length of added together make twice the length of
Verification:
By measurement we find that
PQ = 7.8 cm = 3.9 cm + 3.9 cm

Q 10.

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 of and . Verify by measurement.

SOLUTION:

Steps of construction :
(i) Draw a line 'l' and take a point X on it.
(ii) Construct such that
length = length of
(iii) Then cut off = length of = 3.4 cm
(iv) Thus the length of
= length of - length of

Verification:
By measurement, we find that length of

Q 11.

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

SOLUTION:

Steps of construction :

(i) Draw whose length is not known.
(ii) Fix the compasses pointer on P and the pencil end on Q. The opening of the compasses now gives the length of .
(iii) Draw any line 'l'. Choose a point A on 'l'. Without changing the compasses setting, place the pointer on A.
(iv) Draw an arc that cuts 'l' at a point, say B. Now, is a copy of .

Q 12.

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

SOLUTION:

Steps of construction :

(i) Draw whose length is not known.
(ii) Fix the compasses pointer on A and the pencil end on B. The opening of the compasses now gives the length of
(iii) Draw any line 'l'. Choose a point P on 'l'. Without changing the compasses setting, place the pointer on P.
(iv) Draw an arc that cuts 'l' at a point R.
(v) Now place the pointer on R and without changing the compasses setting, draw another arc that cuts 'l' at a point Q.
(vi) Thus is the required line segment whose length is twice that of AB.

Q 13.

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

SOLUTION:

Steps of construction :

(i) Draw .
(ii) With M as centre and a convenient radius, draw an arc intersecting at two points C and D.
(iii) With C and D as centres and a radius greater than MC, draw two arcs, which cut each other at P.
(iv) Join PM. Then PM is perpendicular to AB through the point M.

Q 14.

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

SOLUTION:

Steps of construction :

(i) Draw any line segment .
(ii) Place a set-square on such that one arm of its right angle aligns along .
(iii) Place a ruler along the edge opposite to the right angle of the set-square.
(iv) Hold the ruler fixed. Slide the set square along the ruler till the point R touches the other arm of the set square.
(v) Join RM along the edge through R meeting at M. Then RM ┴ PQ.

Q 15.

Draw a line l and a point X on it. Through X, draw a line segment perpendicular to l. Now draw a perpendicular to at Y. (use ruler and compasses)

SOLUTION:

Steps of construction :

(i) Draw a line 'l' and take point X on it.
(ii) With X as centre and a convenient radius, draw an arc intersecting the line 'l' at two points A and B.
(iii) With A and B as centres and a radius greater than XA, draw two arcs, which cut each other at E.
(iv) Join XE and produce it to Y. Then XY is perpendicular to 'l'.
(v) With Y as centre and a convenient radius, draw an arc intersecting XY at two points C and D.
(vi) With C and D as centre and radius greater than YD, draw two arcs which cut each other at F.
(vii) Join YF, then YF is perpendicular to XY at Y.

Q 16.

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

SOLUTION:

Axis of symmetry of line segment will be the perpendicular bisector of . So, draw the perpendicular bisector of AB.
Steps of construction :

(i) Draw = 7.3cm.
(ii) With A as centre and radius more than half of AB, draw two arcs, one on each side of AB.
(iii) With B as a centre and the same radius as in step (ii), draw arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the axis of symmetry of the line segment AB.

Q 17.

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

SOLUTION:

Steps of construction :

(i) Draw = 9.5cm.
(ii) With A as centre and radius more than half of AB, draw two arcs one on each side of AB.
(iii) With B as a centre and the same radius as in step (ii), draw arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the perpendicular bisector of .

Q 18.

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?

SOLUTION:

Steps of construction :

(i) Draw = 10.3cm.
(ii) With X as centre and radius more than half of XY, draw two arcs one on each side of XY.
(iii) With Y as centre and the same radius as in step (ii), draw two arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the required perpendicular bisector of .
Now
(A) Take any point P on the bisector drawn. With the help of divider we can check that if P is the point of intersection of XY and CD.
(B) If M is the mid-point of , then

Q 19.

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

SOLUTION:

Steps of construction :

(i) Draw AB = 12.8 cm.
(ii) Draw the perpendicular bisector of which cuts it at C. Thus, C is the mid-point of .
(iii) Draw the perpendicular bisector of which cuts it at D. Thus D is the mid-point of AC.
(iv) Again, draw the perpendicular bisector of which cuts it at E. Thus, E is the mid-point of .
(v) Now, point C, D and E divide in four equal parts.
(vi) By actual measurement, we find that

Q 20.

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

SOLUTION:

Steps of construction :

(i) Draw a line segment = 6.1cm.
(ii) Draw the perpendicular bisector of PQ which cuts, it at O. Thus O is the mid-point of .
(iii) Taking O as centre and OP or OQ as radius draw a circle where is the diameter.

Q 21.

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.

SOLUTION:

Steps of construction :

(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw any chord .
(iii) With A as center and radius more than half of draw two arcs one on each side of AB.
(iv) With B as a centre and the radius same as in step (iii), draw two arcs cutting the arcs drawn in the previous step at P and Q.
(v) Join PQ. Then PQ is the perpendicular bisector of .
(vi) This perpendicular bisector of passes through the centre C of the circle.

Q 22.

Repeat Question 6, if happens to be a diameter.

SOLUTION:

Steps of construction :

(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw its diameter AB.
(iii) With A as center and radius more than half of AB, draw two arcs one on each side of AB.
(iv) With B as a centre and the radius same as in step (iii), draw two arcs cutting the arcs drawn in the previous step at P and Q.
(v) Join PQ. Then PQ is the perpendicular bisector of .
(vi) We observe that this perpendicular bisector of intersect it at the centre C of the circle.

Q 23.

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

SOLUTION:

Steps of construction :

(i) Draw the circle with centre O and radius 4 cm.
(ii) Draw any two chords AB and CD in this circle.
(iii) With A as center and radius more than half AB, draw two arcs one on each side of AB.
(iv) With B as centre and radius same as in step (ii), draw two arcs cutting the arcs drawn in previous step at E and F.
(v) Join EF. Thus EF is the perpendicular bisector of chord AB.
(vi) Similarly draw GH the perpendicular bisector of chord CD.
(vii) These two perpendicular bisectors meet at O, the centre of the circle.

Q 24.

Draw any angle with vertex O. Take a pointA 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 ?

SOLUTION:

Steps of construction :

(i) Draw any angle with vertex O.
(ii) Take a point A on one of its arms and B on another such that OA = OB.
(iii) Draw perpendicular bisector of OA and OB.
(iv) Let they meet at P. Join PA and PB.
(v) With the help of divider, we check that PA = PB.

Q 25.

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

SOLUTION:

Steps of construction :

(i) Draw a line l and mark a point O on it.
(ii) Place the pointer of the compasses at O and draw an arc of any radius which intersects the line l at Q.
(iii) Taking same radius, with centre Q, draw an arc which cuts the previous arc at B.
(iv) Join OB, then ∠BOQ = 60°.
(v) Taking same radius, with centre B, draw an arc which cuts the arc drawn in step (ii) at C.
(vi) Draw bisector of ∠BOC which cuts the at D. Thus, ∠DOQ = 90°.
(vii) Draw OP as bisector of ∠DOB. Thus, ∠POQ = 75°.

Q 26.

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

SOLUTION:

Steps of construction :

(i) Draw .
(ii) With the help of protractor, draw ∠AOB = 147­°.
(iii) Taking centre O and any convenient radius, draw an arc which intersects at P and Q respectively.
(iv) Taking P as centre and radius more than half of PQ, draw an arc.
(v) Taking Q as centre and with the same radius, draw another arc which intersects the previous arc at R.
(vi) Join OR and produce it.
Thus, is the required bisector of ∠AOB.

Q 27.

Draw a right angle and construct its bisector.

SOLUTION:

Steps of construction :

(i) Draw .
(ii) With the help of protractor draw ∠QOP = 90°
(iii) Taking O as centre and any convenient radius, draw an arc which intersect the at A and B respectively.
(iv) Taking B and A as centre and radius more than half of BA, draw two arcs which intersect each other at the point D.
(v) Join OD. Thus, is the required bisector of ∠QOP.

Q 28.

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

SOLUTION:

Steps of construction :

(i) Draw .
(ii) At O, with the help of a protractor, draw ∠AOB = 153­°.
(iii) Draw as the bisector of ∠AOB.
(iv) Again, draw as the bisector of ∠AOC.
(v) Again, draw as the bisector of ∠BOC.
(vi) Thus, divide ∠AOB in four equal parts.

Q 29.

Construct with ruler and compasses, angles of following measures:
(A) 60°
(B) 30°
(C) 90°
(D) 120°
(E) 45°
(F) 135°

SOLUTION:

Steps of construction :
(A) 60°

(i) Draw .
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at P.
(iii) Taking P as centre and same radius, draw an arc which cut the previous arc at Q.
(iv) Join OQ and produce it to B. Thus, ∠BOA is the required angle of 60°.
(B) 30°

(i) Draw .
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at P.
(iii) Taking P as centre and same radius, draw an arc which cut the previous arc at Q.
(iv) Join OQ and produce it to B. Thus, ∠BOA is the angle of 60°.
(v) Taking P as centre and radius more than half of PQ, draw an arc.
(vi) Taking Q as centre and with same radius, draw an arc which cut the previous arc at C.
(vii) Join OC. Thus ∠COA is the required angle of 30°.
(C) 90°

(i) Draw .
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at X.
(iii) Taking X as centre and same radius, draw an arc which cut the previous arc at Y.
(iv) Taking Y as centre and same radius, draw another arc intersecting the arc drawn in step (ii) at Z.
(v) Taking Y and Z as centres and same radius, draw two arcs intersecting each other at S.
(vi) Join OS and produce it to form a ray OB. Thus, ∠BOA is required angle of 90°.
(D) 120°

(i) Draw .
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at P.
(iii) Taking P as centre and same radius, draw an arc which cut the previous arc at Q.
(iv) Taking Q as centre and same radius draw another arc intersecting the arc drawn in step (ii) at S.
(v) Join OS and produce it to D.
Thus, ∠AOD is the required angle of 120°.
(E) 45°

(i) Draw .
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at X.
(iii) Taking X as centre and same radius, draw an arc which cut the previous arc at Y.
(iv) Taking Y as centre and same radius, draw another arc intersecting the arc drawn in step (ii) at Z.
(v) Taking Y and Z as centres and same radius, draw two arcs intersecting each other at S.
(vi) Join OS and produce it to B. Thus, ∠BOA is the angle of 90°.
(vii) Draw OM the bisector of ∠BOA.
Thus, ∠MOA is the required angle of 45°.
(F) 135°

(i) Draw and take a point O on it.
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at A and B.
(iii) Taking A and B as centres and radius more than half of AB, draw two arcs intersecting each other at R.
(iv) Join OR. Thus, ∠QOR = ∠POR = 90°.
(v) Draw the bisector of ∠POR.
Thus, ∠QOD is the required angle of 135°.

Q 30.

Draw an angle of measure 45° and bisect it.

SOLUTION:

Steps of construction :

(i) Draw a line PQ and take a point O on it.
(ii) Taking O as centre and a convenient radius, draw an arc which intersects at two points A and B.
(iii) Taking A and B as centres and radius more than half of AB, draw two arcs which intersect each other at C.
(iv) Join OC. Then ∠COQ is an angle of 90°.
(v) Draw as the bisector of ∠COQ. Thus ∠QOE = 45°.
(vi) Again draw as the bisector of ∠QOE.

Q 31.

Draw an angle of measure 135° and bisect it.

SOLUTION:

Steps of construction :

(i) Draw a line and take a point O on it.
(ii) Taking O as centre and convenient radius, draw an arc, which intersects at A and B.
(iii) Taking A and B as centres and radius more than half of AB, draw two arcs intersecting each other at R.
(iv) Join OR. Thus, ∠QOR = ∠POR = 90°.
(v) Draw the bisector of ∠POR. Thus, ∠QOD is the required angle of 135°.
(vi) Now, draw as the bisector of ∠QOD.
.

Q 32.

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

SOLUTION:

Steps of construction :

(i) Draw an angle of 70° with the help of protractor, i.e., ∠POQ = 70°.
(ii) Draw .
(iii) Place the compasses at O and draw an arc to cut the rays of ∠POQ at L and M.
(iv) Use the same compasses setting to draw an arc with A as centre, cutting AB at X.
(v) Set your compasses setting to the length LM with the same radius.
(vi) Place the compasses pointer at X and draw the arc to cut the arc drawn earlier at Y.
(vii) Join AY.
Thus, ∠YAX = 70°.

Q 33.

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

SOLUTION:

Steps of construction :

(i) Draw an angle of 40° with the help of protractor, i.e., ∠AOB = 40°.
(ii) Draw a line .
(iii) Take any point M on PQ.
(iv) Place the compasses at O and draw an arc to cut the rays of ∠AOB at l and N.
(v) Use the same compasses setting to draw an arc M as centre, cutting MQ at X.
(vi) Set your compasses to length LN with the same radius.
(vii) Place the compasses at X and draw the arc to cut the arc drawn earlier at Y.
(viii) Join MY.
(ix) Thus, ∠QMY = 40° and ∠PMY is supplementary of it.