Constructions Exercise 17.2

Question: 1

Draw △ABC in which AB = 5.5 cm. BC = 6 cm and CA = 7 cm. Also, draw perpendicular bisector of side BC.

Solution:

Steps of construction:

• Draw a line segment AB of length 5.5 cm.

• From B, cut an arc of radius 6 cm.

• With centre A, draw an arc of radius 7 cm intersecting the previously drawn arc at C.

• Join AC and BC to obtain the desired triangle.

• With centre B and radius more than half of BC, draw two arcs on both sides of BC.

• With centre C and the same radius as in the previous step, draw two arcs intersecting the arcs drawn in the previous step at X and Y.

• Join XY to get the perpendicular bisector of BC.
   

Question: 2

Draw ∆PQR in which PQ = 3 cm, QR. 4 cm and RP = 5 cm. Also, draw the bisector of ∠Q

Solution:

Steps of construction:

• Draw a line segment PQ of length 3 cm.

• With Q as centre and radius 4 cm, draw an arc.

• With P as centre and radius 5 cm, draw an arc intersecting the previously drawn arc at R.

• Join PR and OR to obtain the required triangle.

• From Q, cut arcs of equal radius intersecting PQ and QR at M and N, respectively.

• From M and N, cut arcs of equal radius intersecting at point S.

• Join QS and extend to produce the angle bisector of angle PQR.

• Verify that angle PQS and angle SQR are equal to 45° each.
   

Question: 3

Draw an equilateral triangle one of whose sides is of length 7 cm.

Solution:

Steps of construction:

• Draw a line segment AB of length 7 cm.

• With centre A, draw an arc of radius 7 cm.

• With centre B, draw an arc of radius 7 cm intersecting the previously drawn arc at C.

• Join AC and BC to get the required triangle.
   

Question: 4

Draw a triangle whose sides are of lengths 4 cm, 5 cm and 7 cm. Draw the perpendicular bisector of the largest side.

Solution:

Steps of construction:

Draw a line segment PR of length 7 cm.

• With centre P, draw an arc of radius 5 cm.

• With centre R, draw an arc of radius 4 cm intersecting the previously drawn arc at Q.

• Join PQ and QR to obtain the required triangle.

• From P, draw arcs with radius more than half of PR on either sides.

• With the same radius as in the previous step, draw arcs from R on either sides of PR intersecting the arcs drawn in the previous step at M and N.

• MN is the required perpendicular bisector of the largest side.
   

Question: 5

Draw a triangle ABC with AB = 6 cm, BC = 7 cm and CA = 8 cm. Using ruler and compass alone, draw (i) the bisector AD of ∠A and (ii) perpendicular AL from A on BC. Measure LAD.

Solution:

Steps of construction:

Draw a line segment BC of length 7 cm.

With centre B, draw an arc of radius 6 cm.

With centre C, draw an arc of radius 8 cm intersecting the previously drawn arc at A.

Join AC and BC to get the required triangle.

Angle bisector steps:

• From A, cut arcs of equal radius intersecting AB and AC at E and F, respectively.

• From E and F, cut arcs of equal radius intersecting at point H.

• Join AH and extend to produce the angle bisector of angle A, meeting line BC at D.

Perpendicular from Point A to line BC steps:

• From A, cut arcs of equal radius intersecting BC at P and Q, respectively (Extend BC to draw these arcs).

• From P and Q, cut arcs of equal radius intersecting at M.

• Join AM cutting BC at L.

• AL is the perpendicular to the line BC.

• Angle LAD is 15°.
   

Question: 6

Draw △DEF such that DE= DF= 4 cm and EF = 6 cm. Measure ∠E  and ∠F.

Solution:

Steps of construction:

• Draw a line segment EF of length 6 cm.

• With E as centre, draw an arc of radius 4 cm.

• With F as centre, draw an arc of radius 4 cm intersecting the previous arc at D.

• Join DE and DF to get the desired triangle DEF.

• By measuring we get, ∠E= ∠F= 40°..
   

Question: 7

Draw any triangle ABC. Bisect side AB at D. Through D, draw a line parallel to BC, meeting AC in E. Measure AE and EC.

Solution:

Steps of construction:

We first draw a triangle ABC with each side = 6 cm.

Steps to bisect line AB:

• Draw an arc from A on either side of line AB.

• With the same radius as in the previous step, draw an arc from B on either side of AB intersecting the arcs drawn in the previous step at P and Q.

• Join PQ cutting AB at D. PQ is the perpendicular bisector of AB.

Parallel line to BC:

• With B as centre, draw an arc cutting BC and BA at M and N, respectively.

• With centre D and the same radius as in the previous step, draw an arc on the opposite side of AB to cut AB at Y.

• With centre Y and radius equal to MN, draw an arc cutting the arc drawn in the previous step at X.

• Join XD and extend it to intersect AC at E.

• DE is the required parallel line.