**Important Questions & Solutions For Class 9 Maths Chapter 11 (Constructions)**

**Q.1: Construct an equilateral triangle, given its side and justify the construction.**

Solution:

Construction Procedure:

- Let draw a line segment AB=4 cm .
- With A and B as centres, draw two arcs on the line segment AB and note the point as D and E.
- With D and E as centres, draw the arcs that intersect the previous arc respectively that forms an angle of 60Â° each.
- Now, draw the lines from A and B that are extended to meet each other at point C.
- Therefore, ABC is the required triangle.

**Justification:**

From construction, it is observed that

AB = 4 cm, âˆ A = 60Â° and âˆ B = 60Â°

We know that, the sum of the interior angles of a triangle is equal to 180Â°

âˆ A + âˆ B + âˆ C = 180Â°

SuBstitute the values

â‡’ 60Â° + 60Â° + âˆ C = 180Â°

â‡’ 120Â° + âˆ C = 180Â°

â‡’ âˆ C = 60Â°

While measuring the sides, we get

BC = CA = 4 cm (Sides opposite to equal angles are equal)

AB = BC = CA = 4 cm

âˆ A = âˆ B = âˆ C = 60Â°

Hence, justified.

**Q.2: Construct a triangle ABC in which BC = 7cm, âˆ B = 75Â° and AB + AC = 13 cm.**

Solution:

Construction Procedure:

The steps to draw the triangle of given measurement is as follows:

- Draw a line segment of base BC = 7 cm
- Measure and draw âˆ B = 75Â° and draw the ray BX
- Take a compass and measure AB + AC = 13 cm.
- With B as a centre and draw an arc at the point be D
- Join DC
- Now draw the perpendicular bisector of the line BD and the intersection point is taken as A.
- Now join AC
- Therefore, ABC is the required triangle.

**Q.3: Construct a triangle XYZ in which âˆ Y = 30Â°, âˆ Z = 90Â° and XY + YZ + ZX = 11 cm.**

Solution:

**Construction Procedure:**

The steps to draw the triangle of given measurement is as follows:

- Draw a line segment AB which is equal to XY + YZ + ZX = 11 cm.
- Make an angle âˆ Y = 30Â° from point A and the angle be âˆ LAB
- Make an angle âˆ Z = 90Â° from point B and the angle be âˆ MAB
- Bisect âˆ LAB and âˆ MAB at point X.
- Now take the perpendicular bisector of the line XA and XB and the intersection point be Y and Z respectively.
- Join XY and XZ
- Therefore, XYZ is the required triangle

**Q.4: Construct a triangle ABC in which BC = 8cm, âˆ B = 45Â° and AB â€“ AC = 3.5 cm.**

Solution:

Construction Procedure:

The steps to draw the triangle of given measurement is as follows:

- Draw a line segment of base BC = 8 cm
- Measure and draw âˆ B = 45Â° and draw the ray BX
- Take a compass and measure AB â€“ AC = 3.5 cm.
- With B as centre and draw an arc at the point be D on the ray BX
- Join DC
- Now draw the perpendicular bisector of the line CD and the intersection point is taken as A.
- Now join AC
- Therefore, ABC is the required triangle.

**Q.5: Draw a line segment AB of 4 cm in length. Draw a line perpendicular to AB through A and B, respectively. Are these lines parallel?**

Solution:

According to the question,

A line segment AB of length 4cm.

To draw aÂ perpendicularÂ to AB through A and B, respectively.

Steps of construction:

1. Draw AB = 4 cm.

2. With A as centre, draw an arc, intersecting AB at P.

3. With P as centre and the same radius, draw an arc intersecting the arc drawn in step 2 at Q.

4. With Q as centre and the same radius, draw an arc, intersecting the arc drawn in step 3 at R.

5. With R as centre and the same radius, draw an arc, intersecting the arc drawn in step 5 at X.

6. Draw OX and produced it to C and D.

7. Now, repeat the steps from 2 to 7 to draw the line EF perpendicular through B.

Yes, these lines are parallel because the sum of the interior angles on the same side of the transversal is 180 degrees.

### Extra Questions To Practice For Class 9 Maths Chapter 11 (CBSE)

- Construct a triangle PQR in which QR = 6cm, âˆ Q = 60Â° and PR â€“ PQ = 2cm.
- Construct a right triangle whose base is 12cm and sum of its hypotenuse and other side is 18 cm.
- Draw an angle of 80Â° with the help of a protractor. Then construct angles of
- (i) 40Â°
- (ii)160Â°
- (iii) 120Â°

- Construct a triangle whose sides are 3.6 cm, 3.0 cm and 4.8 cm. Bisect the smallest angle and measure each part.
- A triangle if its perimeter is 10.4 cm and two angles are 45Â° and 120Â°.