RD Sharma Solutions Class 6 Maths Chapter 19 Geometrical Constructions

Exercise 19.1

Question 1: Construct line segments whose lengths are:

(i) 4.8 cm

(ii) 12 cm 5 mm

(iii) 7.6 cm 

Solution 1:

(i) 4.8 cm

1. Make a line name it l and mark point A on it.

2. Now place the metal point of the compass at 0 position mark of the ruler.

3. Open the compass such that the pencil point is at 4.8 cm mark on the ruler.

4. Take compass on L such that its metal point is on A.

5.Now mark a small mark as point B on L which is corresponding to the pencil point of the compass.

6. Here, a line segment of length 4.8 cm namely AB. 

(ii) 12 cm 5 mm

1. Make a line name it l and mark point A on it.
2. Now place the compass at 0 position mark of the ruler.
3. Open the compass such that the pencil point is at 5 small points from the mark of 12 cm to 13 cm on the ruler.
4. Take compass on L such that its metal point is on A.
5. Now mark a small mark as B on L which is corresponding to the pencil point of the compass.
6. Here, a line segment of length 12 cm 5 mm namely AB. 

(iii) 7.6 cm

1. Make a line name it l and mark point A on it.
2. Now place the compass at 0 position mark of the ruler.
3. Open the compass such that the pencil point is at 6 small points from the mark of 7 cm to 8 cm of the ruler.
4. Take compass on L such that its metal point is on A.
5. Now mark a small mark as B on L which is corresponding to the pencil point of the compass.
6. Here, a line segment of length 7.6 cm namely AB. 

Question 2: Construct two segments of length 4.3 cm and 3.2 cm. Construct a segment whose length is equal to the sum of the lengths of these segments.

Solution 2:

1.  Make AB and CD of lengths 4.3 cm and 3.2 cm with the help of compass and ruler.
2. Construct a line L and mark point P on it.
3. Now place the compass at zero mark of the ruler.
4. Make adjustments in the compass such that the pencil point reaches the point B.
5. Take compass on L such that its metal point is on P.
6. Now make a small mark as Q on the line L corresponding to the pencil point of the compass.
7. Reset the compass so that its metal and pencil points are on C and D.
8. Take compass on L such that its metal point is on Q and the pencil point makes a small mark as R which is opposite to P on L.

Here, PR is the line segment whose length is equal to the sum of the  3.2cm + 4.3 cm = 7.5 cm.

Exercise 19.2

Question 1:  How many lines can be drawn which are perpendicular to a given line and pass through a given point lying

(i) outside it?

(ii) on it? 

Solution1:

(i) A perpendicular line from a given point to a given line is the shortest distance between them.

only one shortest distance is possible here.

Here, only one perpendicular line is possible from a given point lying outside it.

(ii) from any point on the line, we can draw only one perpendicular line.

Here, only one perpendicular line can be drawn from a given point lying on it.

Question 2: Draw a line PQ. Take a point R on it. Draw a line perpendicular to PQ and passing through R.

(Using (i) ruler and a set-square (ii) ruler and compasses) 

Solution 2:   Using ruler and a set- square:

1. Make a line PQ and mark a point R on it.
2. Place the set square with its one arm of the right angle (90 deg.) along the line PQ.
3. Place the ruler along its edge without disturbing the position of set square.
4. Remove the set square without disturbing the position of the ruler and construct a line MN through R.
5. Here, MN is the required line which is perpendicular to PQ through the point R.

Using ruler and compasses:

1.  Construct a line PQ and mark R on it.
2. Take R as the center and measuring convenient radius, draw an arc which touches the line PQ at A and B.
3. Take A and B as centers and radius which is greater than AR, draw two arcs which cuts each other at the point of S.
4. Now join the points RS and extend in both directions.

Here, RS is the required line which is perpendicular to PQ through the point R.

Question 3:  Draw a line l. Take a point A, not lying on l. Draw a line m such that m ⊥ l and passing through A.

(Using (i) ruler and a set-square (ii) ruler and compasses) 

Solution 3:

using ruler and a set- square:

1. Make a line L and mark a point A outside it.
2. Now place the set square PQR with its one arm PQ of the right angle along the line L.
3. Place the ruler along the edge PR without disturbing the position of set square.
4. Slide the set square along the ruler until its arm QR reaches A without disturbing the position of the ruler. Draw a line m.
5. Here, line m is the required line which is perpendicular to the line L.

Using ruler and compasses:

1. As, A as center, make an arc PQ which intersects the line L at P and Q.
2. By taking P and Q as centers, draw two arcs which intersects each other at the point B.
3. Now connect the points A and B and extend in both the directions.
4. Here, AB is the required line.

Question 4. Draw a line AB and take two points C and E on opposite sides of AB. Through C, draw CD ⊥ AB and through E draw EF ⊥ AB.

(Using (i) ruler and a set-square (ii) ruler and compasses) 

Solution 4:

using ruler and a set- square:

1. Make a line AB and mark two points C and E on the opposite sides of AB.
2. Put the set square PQR on the side E with its one arm PQ of the right angle along the line AB.
3. Put the ruler along the edge PR.
4. Slide the set square along the ruler until the arm QR reaches C.
5. Make a line CD where D is a point on the line AB.
6. Here CD is the required line where CD ⊥ AB.
7. Now follow the same steps using a set square on the side E, we construct a line EF ⊥ AB.

Using ruler and compasses:

1. Make a line AB and mark two points C and E on the opposite sides.
2. Consider C as center, construct an arc PQ which intersects AB at P and Q.
3. Consider P and Q as centers, draw two arcs which intersect each other at the point H.
4. Now connect the points H and C where HC crosses the line AB at the point D.

As, CD ⊥ AB

1. In the same way, consider E as center and construct an arc RS.
2. Consider R and S as centers, construct two arcs which intersect each other at the point G.
3. Now join the point G and E such that GE crosses the line AB at the point F

As, EF ⊥ AB.

Question 5:   Draw a line segment AB of length 10 cm. Mark a point P on AB such that AP = 4 cm. Draw a line through P perpendicular to AB. 

Solution 5:

1. Make a line L and mark a point A on it.
2. With the help of a ruler and a compass mark a point B which is at 10 cm from the point A on the line L.
3. Here, AB is the required line segment of length 10 cm.
4. Now, mark a point P which is 4 cm from the point A in the direction of B.
5. Taking P as center and radius 4 cm draw an arc which intersects the line L at A and E.
6. Taking A and E as centers and radius 6 cm draw two arcs which intersects each other at the point R.
7. Now join P and R and extend it.
8. Here, PR is the required line which is perpendicular to the line AB.

Question 7:  Using a protractor, draw ∠BAC of measure 70^o. On side AC, take a point P, such that AP = 2 cm. From P draw a line perpendicular to AB. 

Solution 7:   Construct a line segment AC on the line L

1.  Put a protractor on the line segment AC such that it coincides with the line of diameter of protractor and the middle of the line coincides with A.
2.  Now by counting from the right side, mark B as the point of 70^o of the protractor and construct AB.
3. Measure 2 cm from the point A on the line segment AC and mark P.
4. Consider P as center, construct an arc which intersects the line l at E and F.
5. Consider the same radius and the points E and F as centers, draw two arcs which intersects G on the other side.
6. Connect the points P and G.

Question 8. Draw a line segment AB of length 8 cm. At each end of this line segment, draw a line perpendicular to AB. Are these two lines parallel? 

Solution 8:

1. By taking a convenient radius with A as center make  an arc which intersects the line at W and X.
2.  Taking W and X as centers and radius which is more than AW, draw two arcs which intersects each other at the point M.
3. Now connect the points AM and extend in both directions to P and Q.
4. With B as center and convenient radius make an arc which intersects the line at Y and Z.
5.  Taking Y and Z as centers and radius which is greater than YB, draw two arcs which intersects each other at the point N.
6. Now connect the points BN and extend in both directions at the point S and R.
7. Consider the lines perpendicular to the points A and B be PQ and RS

We know that ∠QAB = ∠ABR = 90^o

1. Two alternate interior angles are equal when the two parallel lines are intersected by a third line.
2. Hence, PQ and RS are parallel.

Question 9.  Using a protractor, draw ∠BAC of measure 45^o. Take a point P in the interior of ∠BAC. From P drawn line segments PM and PN such that PM ⊥ AB and PN ⊥ AC, Measure ∠MPN. 

Solution 9:

1.  Make a line segment A on the line L.
2. Put the protractor on the line segment AC such that it coincides with the line of the diameter of the protractor and the middle point of the line coincides with A.
3. By counting from the right side, mark B as the point of 45^o of protractor and construct AB.
4. Taking P as center and convenient radius draw an arc which intersects the line segment AB at points T and Q and AC at point R and S.
5. Taking T and Q as centers and same radius, draw two arcs which intersects at point G on the other side.
6. Taking R and S as centers and same radius, draw two arcs which intersects at point H on the other side.
7. Now join the points PG and PH which intersects the line segments AB and AC at points M and N.
8. By measuring ∠MPN with the help of a protractor we get ∠MPN = 135^o.

Question 4:  Draw a circle with center at point O. Draw its two chords AB and CD such that AB is not parallel to CD. Draw the perpendicular bisectors of AB and CD. At what point do they intersect? 

Solution 4:   Make a circle with O as the center. Construct two chords AB and CD.

1.  Taking A as center and radius more than half of AB cut an arc on both sides of the line segment AB.
2. Taking B as center and same radius, cut an arc which cuts the previous arcs at the points P and Q.
3. Now join the points P and Q.
4. Taking C as center and radius more than half of CD construct arcs on both sides of CD.
5. Taking D as center and same radius, construct arcs which cuts the previous arcs at the points R and S.
6. Now join the points R and S.
7. Make the line segments of perpendicular bisector of AB and CD.
8. Here, the perpendicular bisector of line segments AB and CD meet at the point O which is the center of the circle.

 

Question 6. Draw a line segment AB and bisect it. Bisect one of the equal parts to obtain a line segment of length 1/2 (AB). 

Solution 6:  Make a line segment AB.

1. Taking A as center and radius which is more than half of AB, cut an arc on both sides of AB.
2. Taking B as center and same radius, cut an arc which cuts the previous arcs at the points P and Q.
3. Now join the points P and Q such that the line PQ intersects AB at the point C.
4. Taking A as center and radius more than half of AC, cut an arc on both sides of AC.
5. Taking C as center and same radius cut an arc which cuts the previous arcs at the points R and S.
6. Now join the points R and S such that the line intersects AB at the point D.

Here, AC = CB = 1/2 (AB)

By dividing AC at D

As we know that AD and AC are of same length

Hence, AD = AC = 1/4 (AB)

Question 7. Draw a line segment AB and by ruler and compasses, obtain a line segment of length 3/4 (AB). 

Solution 7:   Make a line segment AB with the help of a ruler.

1. Taking A as center and radius which is more than half of AB, cut an arc on both sides of AB.
2. Taking B as center and same radius construct arcs which cuts the previous arcs at the points P and Q.
3. Now join the points P and Q such that the line intersects AB at the point C.
4. Taking A as center and radius which is more than half of AB construct arcs on both sides of AC.
5. Taking C as center and same radius construct arcs which cuts the previous arcs at the points R and S.
6. Now join the points R and S such that the line intersects AB at the point D.

Bisect AC and mark D as the point of bisection.

As,we know that

AD = 1/4 (AB)

It can be written as

DC = 1/4 (AB) and CB = 1/2 (AB)

So, we get

DB = 1/4 (AB) + 1/2 (AB) = 3/4 (AB)

Hence, DB is the required line segment of length 3/4 (AB).

Exercise 19.4

Question 1. Construct the following angles with the help of a protractor:

45^o, 67^o, 38^o, 110^o, 179^o, 98^o, 84^o 

Solution 1:

Draw a ray OA, Now place the protractor on the ray OA such that its center coincides with point O and the diameter coincides with OA. Mark a point B against the mark of 45^o on the protractor.

Remove the protractor and construct the line OB where ∠AOB is the required angle.

Use the same way for draw the angles 67^o, 38^o, 110^o, 179^o, 98^o and 84^o

Question 2: Draw two rays PQ and RS as shown in Fig. 19.14 (i) and (ii). Using protractor, construct angles of 15^o and 138^o with one arm PQ and RS respectively.

Solution 2:

(i) Construct a ray PQ:

1. Place the protractor on PQ such that its center coincides with the point P and the diameter coincides with PQ.

2. Mark B against the mark of 15^o on the protractor.

3. Construct PB by removing the protractor.

Here ∠QPB is the required angle of 15^o.

(ii) Construct a ray RS :

1. Place the protractor on RS such that its center coincides with the point R and the diameter coincides with RS.
2. Mark T against the mark of 138^o on the protractor.
3. Construct ST by removing the protractor.

Exercise 19.5

Question 1. Draw an angle and label it as ∠BAC. Construct another angle, equal to ∠BAC. 

Solution 1:

Question 3. Using your protractor, draw an angle of measure 108^o. With this angle as given, drawn an angle of 54^o. 

Solution 3:  Construct a ray OA.

1. By using protractor, draw an angle ∠AOB = 108^o where 108/2 = 54^o

As we know, 54^o is half of 108^o.

1. In order to get angle 54^o, we must bisect the angle of 108^o.
2. Taking O as center and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q.
3. Taking P as center and radius which is more than half of PQ construct an arc.
4. Taking Q as center and same radius construct another arc which intersects the previous arc and name it as point R.
5. Now join the points O and R and extend it to the point X.

Here, ∠AOX is the required angle of 54^o.

Question 4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45^o. 

Solution 4:   Construct a ray OA.

By using a protractor construct ∠AOB of 90^o.

1. Taking O as center and convenient radius, cut an arc which cuts the sides OA and OB at the points P and Q.
2. Taking P as center and radius which is more than half of PQ, cut an arc.
3. Taking Q as center and same radius, cut an another arc which intersects the previous arc and name it as point R.
4. Now join the points O and R and extend it to the point X.

Here, ∠AOX is the required angle of 45^o where ∠AOB = 90^o and ∠AOX = 45^o.

Question 5:  Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other. 

Solution 5:   As we know that the two angles which are adjacent and supplementary are known as linear pair of angles.

Make a line AB and mark a point O on it.

By constructing an angle ∠AOC we get another angle ∠BOC.

Now bisect ∠AOC using a compass and a ruler and get the ray OX.

In the same way bisect ∠BOC and get the ray OY.

∠XOY = ∠XOC + ∠COY

It can be written as

∠XOY = 1/2 ∠AOC + 1/2 ∠BOC

So we get

∠XOY = 1/2 (∠AOC + ∠BOC)

∠AOC and ∠BOC are supplementary angles

∠XOY = 1/2 (180) = 90^o

Question 6: Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line. 

Solution 6:  Make two lines AB and CD which intersects each other at the point O

Since vertically opposite angles are equal, we get

∠BOC = ∠AOD and ∠AOC = ∠BOD

Now bisect angle AOC and construct the bisecting ray as OX.

In the same way, we bisect ∠BOD and construct bisecting ray OY.

We get

∠XOA + ∠AOD + ∠DOY = 1/2 ∠AOC + ∠AOD + 1/2 ∠BOD

∠AOC = ∠BOD

∠XOA + ∠AOD + ∠DOY = 1/2 ∠BOD + ∠AOD + 1/2 ∠BOD

So we get

∠XOA + ∠AOD + ∠DOY = ∠AOD + ∠BOD

AB is a line

∠AOD and ∠BOD are supplementary angles whose sum is equal to 180^o.

∠XOA + ∠AOD + ∠DOY = 180^o

The angles on one side of a straight line is always 180^o and also the sum of angles is 180^o

Here, XY is a straight line where OX and OY are in the same line.

Question 7. Using ruler and compasses only, draw a right angle. 

Solution 7:  Draw a ray OA.

1. Taking O as center and convenient radius cut an arc PQ using a compass intersecting the ray OA at the point Q.
2. Taking P as center and same radius cut an arc which intersects the arc PQ at the point R.
3. Taking R as center and same radius, cut an arc which cuts the arc PQ at the point C opposite to P.
4. Using C and R as the center cut two arc of radius which is more than half of CR intersecting each other at the point S.
5. Now join the points O and S and extend it to the point B.

Here, ∠AOB is the required angle of 90^o.

Question 8. Using ruler and compasses only, draw an angle of measure 135^o. 

Solution 8:  Draw a line AB and mark a point O on it.

1. Taking O as center and convenient radius, cut an arc PQ using a compass which intersects the line AB at the point P and Q.2
2. Taking P as center and same radius, construct another arc which intersects the arc PQ at the point R.
3. Taking Q as center and same radius, construct another arc which intersects the arc PQ at the point S which is opposite to P.
4. Considering S and R as centers and radius which is more than half of SR, construct two arcs which intersects each other at the point T.
5. Now join the points O and T which intersects the arc PQ at the point C.
6. Considering C and Q as centers and radius which is more than half of CQ, construct two arcs which intersects each other at the point D.
7. Now join the points O and D and extend it to point X to form the ray OX.

Here, ∠AOX is the required angle of 135^o.

Question 9: Using a protractor, draw an angle of measure 72^o. With this angle as given, draw angles of measure 36^o and 54^o.  

Exercise 19.6

Question 1.  Construct an angle of 60^o with the help of compasses and bisect it by paper folding. 

Solution 1:   Draw a ray OA

1. Taking O as center and convenient radius, construct an arc which cuts the ray OA at the point P.
2. Taking P as center and same radius, construct another arc which cuts the previous arc at the point Q.
3. Construct OQ and extend it to the point B.
4. Here, ∠AOB is the required angle of 60^o.
5. Now cut the part of paper as sector OPQ.
6. Fold the part of paper such that the line segments OP and OQ coincides.
7. The angle which is made at point O is the required angle which is half of ∠AOB.

Question 2. Construct the following angles with the help of ruler and compasses only:

(i) 30^o

(ii) 90^o

(iii) 45^o

(iv) 135^o

(v) 150^o

(vi) 105^o 

Solution 2:

(i) 30^o

Draw a ray OA.

1. Taking O as center and with convenient radius, construct an arc which cuts OA at the point P.
2. Taking P as center and same radius, construct an arc which cuts the previous arc at the point Q.
3. Considering P and Q as centers and radius which is more than half of PQ construct two arcs which cuts each other and name it as point R.
4. Construct OR and extend it to the point B.
5. Here, ∠AOB is the required angle of 30^o

(ii) 90^o

Draw a ray OA.

1. Taking O as center and with convenient radius, construct an arc which cuts OA at the point P.
2. Taking P as center and same radius, construct an arc which cuts the previous arc at the point Q.
3. Taking Q as center and same radius, construct an arc which cuts the previous arc at the point R.
4. Considering Q and R as centers and radius which is more than half of QR construct two arcs which cuts each other and name it as point S.
5. Construct OS and extend it to the point B from the ray OB.
6. Here, ∠AOB is the required angle of 90^o.

(iv) 135^o

Draw a line AB and mark a point O in the middle of AB.

1. Taking O as center and convenient radius, construct an arc which cuts the line AB at the points P and Q.
2. Construct an angle of 90^o on the ray OB as ∠BOC = 90^o where OC cuts the arc at the point R
3. Taking Q and R as centers and radius which is more than half of QR, construct two arcs which cuts each other and name it as point S.
4. Construct OS and extend it to the point D to form the ray OD.
5. Here, ∠BOD is the required angle of 135^o.

(v) 150^o

Draw a line AB and mark a point O in the middle of AB.

1. Taking O as center and convenient radius construct an arc which cuts the line AB at the points P and Q.
2. Taking Q as center and same radius construct an arc which cuts the previous arc and name it as point R.
3. Taking R as center and same radius construct an arc which cuts the previous arc and name it as point S.
4. Taking P and S as centers and radius which is more than half of PS, construct two arcs which cuts each other and name it as point T.
5. Construct OT and extend it to the point C to form the ray OC.

Here, ∠BOC is the required angle of 150^o.

(vi) 105^o

Construct a ray OA and make ∠AOB = 90^o and ∠AOC = 120^o

Bisect ∠BOC and get the ray OD.

Here, ∠AOD is the required angle of 105^o.

Question 3.  Construct a rectangle whose adjacent sides are 8 cm and 3 cm. 

Solution 3:   Construct a line segment AB of length 8 cm.

Draw ∠BAX = 90^o at A and ∠ABY = 90^o at B

Mark a point D on the ray AX where AD = 3 cm, with the help of compass and ruler

In the same way mark the point C on the ray BY where BC = 3 cm

Construct the line segment CD

Hence, ABCD is the required rectangle.