Construction - Exercise 22

Draw a line segment AB = 5.2 cm. Draw a perpendicular to it from a point P outside line AB by using :

(i) ruler and compasses

(ii) ruler and set square

Answer

(i) ruler and compasses

Steps of construction:

1. Draw a line segment AB = 5.2 cm using a ruler.
2. Mark a point P outside the line AB.
3. With P as centre and any convenient radius, draw an arc cutting AB at two points M and N.
4. With M as centre and radius more than half of MN, draw an arc below AB.
5. With N as centre and the same radius, draw another arc cutting the previous arc at Q.
6. Join P and Q.

PQ ⟂ AB, so PQ is the required perpendicular.

(ii) ruler and set square

Steps of construction:

1. Draw a line segment AB = 5.2 cm.
2. Mark a point P outside AB.
3. Place a ruler along line AB.
4. Place a set square such that one of its edges lies along AB.
5. Slide the set square along the ruler until its perpendicular edge passes through point P.
6. Draw a line PQ along that edge meeting AB at Q.

PQ ⟂ AB, hence required perpendicular is obtained

Draw a line segment AB = 4.8 cm. Take a point P on it such that BP = 2.3 cm. Draw a perpendicular to AB at point P using:

(i) ruler and compasses

(ii) ruler and set square

Answer

(i) ruler and compasses

Steps of construction:

1. Draw a line segment AB = 4.8 cm using a ruler.
2. Mark point P on AB such that BP = 2.3 cm.
3. With P as centre and a convenient radius, draw an arc cutting AB at two points M and N.
4. With M as centre and radius more than half of MN, draw an arc above AB.
5. With N as centre and the same radius, draw another arc cutting the previous arc at Q.
6. Join P and Q.

PQ ⟂ AB, so PQ is the required perpendicular at point P.

(ii) ruler and set square

Steps of construction:

1. Draw a line segment AB = 4.8 cm.
2. Mark point P such that BP = 2.3 cm.
3. Place a ruler along line AB.
4. Place a set square so that one of its edges coincides with AB.
5. Adjust the set square so that its perpendicular edge passes through P.
6. Draw a line PQ along that edge.

PQ ⟂ AB, hence required perpendicular is obtained.

Draw any triangle ABC. Through A, draw a line parallel to BC using :

(i) ruler and compasses

(ii) ruler and set square

Answer

Steps of construction:

(i) ruler and compasses

Steps of construction:

1. Draw any triangle ABC.
2. Take any point Q on BC. Join A and Q.
3. With Q as centre and any convenient radius draw an arc cutting BC at M and AQ at N.
4. With the same radius and A as centre, draw an arc cutting AQ at R.
5. Take radius = MN and with R as centre, draw an arc cutting the existing arc at S. Join A and S.

(ii) ruler and set square

1. Draw any triangle ABC.
2. Place a set square such that one of its shorter edges lies along line BC.
3. Place a ruler vertically such that one of its edges just lies along and touches the standing edge of the set square.
4. Hold the ruler firmly and slide the set square along it until its horizontal edge just passes through point A.
5. Draw the line through A along the edge of the set square.

Construct a △ABC such that :

(i) AB = 5 cm, BC = 4.6 cm and AC = 4.3 cm

(ii) BC = 6 cm, CA = 4 cm and AB = 3.5 cm

(iii) CA = 4.8 cm, CB = 4.2 cm and AB = 4.5 cm

Answer

(i) AB = 5 cm, BC = 4.6 cm and AC = 4.3 cm

Steps of construction:

1. Draw AB = 5 cm.
2. With A as centre and radius 4.3 cm (AC), draw an arc.
3. With B as centre and radius 4.6 cm (BC), draw another arc cutting the first arc at C.
4. Join AC and BC.

Then, △ABC is the required triangle.

(ii) BC = 6 cm, CA = 4 cm and AB = 3.5 cm

Steps of construction:

1. Draw BC = 6 cm.
2. With B as centre and radius 3.5 cm (AB), draw an arc.
3. With C as centre and radius 4 cm (CA), draw another arc cutting the first arc at A.
4. Join AB and AC.

Then, △ABC is the required triangle.

(iii) CA = 4.8 cm, CB = 4.2 cm and AB = 4.5 cm

Steps of construction:

1. Draw CA = 4.8 cm.
2. With C as centre and radius 4.2 cm (CB), draw an arc.
3. With A as centre and radius 4.5 cm (AB), draw another arc cutting the first arc at B.
4. Join AB and BC.

Then, △ABC is the required triangle.

Construct a △ABC such that :

(i) BC = 5 cm, ∠BCA = 60° and CA = 4.3 cm.

(ii) AB = 3.7 cm, ∠A = 45° and AC = 5.3 cm.

Answer

(i) BC = 5 cm, ∠BCA = 60° and CA = 4.3 cm.

Steps of Construction:

1. Draw BC = 5 cm.
2. At point C, construct an angle of 60°.
3. With C as centre and radius 4.3 cm, draw an arc cutting the ray CX at A.
4. Join A to B.

Then, △ABC is the required triangle.

(ii) AB = 3.7 cm, ∠A = 45° and AC = 5.3 cm.

Steps of Construction:

1. Draw AB = 3.7 cm.
2. At point A, construct an angle of 45°.
3. With A as centre and radius 5.3 cm, draw an arc cutting the ray AX at C.
4. Join B to C.

Then, △ABC is the required triangle.

Construct a △XYZ such that :

(i) XY = 4.5 cm, ∠X = 60° and ∠Y = 45°.

(ii) YZ = 5 cm, ∠ZYX = 30° and ∠YZX = 75°.

Answer

(i) XY = 4.5 cm, ∠X = 60° and ∠Y = 45°.

Steps of Construction:

1. Draw a line segment XY = 4.5 cm.
2. At point X, construct an angle of 60°.
3. At point Y, construct an angle of 45°.
4. Let the two rays intersect at point Z.
5. Join ZX and ZY.

Then, △XYZ is the required triangle.

(ii) YZ = 5 cm, ∠ZYX = 30° and ∠YZX = 75°.

Steps of Construction:

1. Draw a line segment YZ = 5 cm.
2. At point Y, construct an angle of 30°.
3. At point Z, construct an angle of 75°.
4. Let the two rays intersect at point X.
5. Join XY and XZ.

Then, △XYZ is the required triangle.

Construct an isosceles △ABC such that :

(i) base AB = 4.2 cm, base angle = 30°.

(ii) base AC = 4.5 cm, base angle = 75°.

Answer

(i) base AB = 4.2 cm, base angle = 30°.

Steps of Construction:

1. Draw a line segment AB = 4.2 cm.
2. At point A, construct an angle of 30°.
3. At point B, construct an angle of 30°.
4. Let the two rays intersect at point C.
5. Join AC and BC.

Then, △ABC is the required isosceles triangle.

(ii) base AC = 4.5 cm, base angle = 75°.

Steps of Construction:

1. Draw a line segment AC = 4.5 cm.
2. At point A, construct an angle of 75°.
3. At point C, construct an angle of 75°.
4. Let the two rays intersect at point B.
5. Join AB and BC.

Then, △ABC is the required isosceles triangle.

Construct an isosceles △ABC such that AB = AC = 5 cm and ∠A = 60°.

Answer

Steps of Construction:

1. Draw a line segment AB = 5 cm.
2. At point A, construct an angle of 60°.
3. On this ray, mark point C such that AC = 5 cm.
4. Join BC.

Then, △ABC is the required isosceles triangle.

Construct an isosceles △ABC such that AB = BC = 5.4 cm and ∠B = 30°.

Answer

Steps of Construction:

1. Draw a line segment AB = 5.4 cm.
2. At point B, construct an angle of 30°.
3. On this ray, mark point C such that BC = 5.4 cm.
4. Join AC.

Then, △ABC is the required isosceles triangle.

Construct an equilateral △ABC in which BC = 5 cm.

Answer

Steps of Construction:

1. Draw a line segment BC = 5 cm.
2. With B as centre and radius 5 cm, draw an arc.
3. With C as centre and radius 5 cm, draw another arc cutting the first arc at A.
4. Join AB and AC.

Then, △ABC is the required equilateral triangle.

Construct an equilateral △ABC such that AB = 4.2 cm.

Answer

Steps of Construction:

1. Draw a line segment AB = 4.2 cm.
2. With A as centre and radius 4.2 cm, draw an arc.
3. With B as centre and radius 4.2 cm, draw another arc cutting the first arc at C.
4. Join AC and BC.

Then, △ABC is the required equilateral triangle.

Construct a right angled △ABC in which ∠B = 90°, BC = 4 cm and hypotenuse CA = 5 cm.

Answer

Steps of Construction:

1. Draw a line segment BC = 4 cm.
2. At point B, construct ∠CBX = 90°.
3. With C as centre and radius 5 cm, draw an arc cutting the perpendicular at A.
4. Join AC.

Then, △ABC is the required triangle.

Construct a right angled △ABC in which AB = 4.2 cm, BC = 4.5 cm and ∠B = 90°.

Answer

Steps of Construction:

1. Draw a line segment BC = 4.5 cm.
2. At point B, construct ∠CBX = 90°.
3. On the perpendicular line, mark point A such that AB = 4.2 cm.
4. Join AC.

Then, △ABC is the required triangle.