# Constructions – Exercise 11.1 – (MATHEMATICS) – 9th Class

**Exercise 11.1 **

**1. Construct an angle of 90° at the initial point of a given ray and justify the construction.**

### Show Answer

**Sol. Steps of construction:
**

(i) Let the given ray be OA with initial point 0.

(ii) Taking 0 as centre and any suitable radius, draw an arc to intersect the ray at P.

(iii) Taking P as centreand the same radius, draw an arc to intersect the arc drawn in step (ii) at A_{1}.

(iv) Taking A_{1} as centre and the same radius, draw an arc to cut the arc drawn in step (ii) at A2.

(v) Taking A_{1} and A_{2} as centres and radius greater than ,draw two arcs to intersect each other at L on the same side of the line segment A_{1}A_{2}.

(vi) Join OL and produce it along OL.

Hence, ∠LOA = 90°.

**Justification:** ∠A_{1}OA = 60°, and ∠A_{2}OA_{1} = 60°.

Bisector of ∠A_{2}OA_{1} = ∠LOA_{1}, = = 30°.

Then ∠LOA = ∠LOA_{1} + ∠A_{1}OA = 30° + 60° = 90°.

**2. Construct an angle of 45° at the initial point of a given ray and justify the construction.**

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**Sol. Steps of construction:**

(a) Draw an angle AOP = 90° as in Ans.1.

(b) Taking A and P as centres and radius greater than draw two arcs to intersect each other at B as shown in the adjoining figure.

(c) Join OB and produce it. Hence, acute ∠BOX = 45°.

**Justification:** ∠AOP = 90^{0}

∠AOB + ∠BOP = 90° [ ∠AOB = ∠BOP]

2∠BOP = 90°

∠BOP = 45°.

**3. Construct the angles of the following measurements:**

(i) 30° (ii) (iii) 15°.

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**Sol.**

**(i) Steps of construction:**

(a) Draw a ray OX with initial point O.

(b) Taking O as centre and suitable radius, draw an arc to cut OX at P.

(c) Taking P as centre and same radius draw an arc to cut the arc drawn in step (b) at A.

(d) Taking P and A as centres and radius greater than draw two arcs to intersect each other at A_{1}.

(e) Join OA_{1} and produce it along OA_{1}. Hence, acute angle A_{1},OX = 30°.

(ii) **Steps of construction:**

(a) Draw an angle of 45° on a ray OX such that ∠AOX = 45° as in Q. 2.

(b) Taking B and P as centres and radius greater than , draw arcs to intersect each other at C.

(c) Join OC and produce it to Y. Hence, acute

**(iii) Steps of construction:**

(a) Draw an ∠A_{1}OX = 30° on a ray OX as in Part (i).

(b) Taking A_{2} and P as centres and radius greater than , draw two arcs to intersect each other at A_{3}.

(c) Join 0A_{3} and produce it to B. Hence, acute angle BOX = 15°.

**4. Construct the following angles and verify by measuring them by a protractor.**

(i) 75° (ii) 105° (iii) 135°.

### Show Answer

**Sol. (1) Steps of construction:**

(a) Draw a ray OX with initial point 0.

(b) Taking 0 as centre and suitable radius, draw an arc to intersect OX at P.

(c) Taking P as centre and same radius, draw an arc to intersect the arc drawn in step (b) at A.

(d) Taking A as centre and same radius, draw an arc to intersect the arc drawn in step (b) at B.

(e) Taking A and B as centres and radius greater than draw two arcs to intersect each other at C.

(f) Join OC to intersect the arc drawn in step (b) at D.

(g) Taking A and D as centres and radius greater than draw two arcs to intersect each other at E.

(h) Join OE and produce it to M.

Hence, acute angle ∠MOX = 75°.

**Verification:**

On measuring by a protractor, we get ∠MOX = 75°.

**(ii) Steps of construction:**

(a) Draw a ray OX with initial point 0.

(b) Taking 0 as centre and any suitable radius, draw an arc to intersect OX at P.

(c) Taking P as centre and same radius, draw an arc to intersect the arc drawn in step (b) at A.

(d) Taking A as centre and same radius, draw an arc to intersect the arc in step (h) at B.

(e) Taking A and B as centres and radius greater than , draw two arcs to intersect each other at C.

(f) Join OC to intersect the arc drawn in step (b) at D.

(g) Taking B and D as centres and any suitable radius greater than , draw two arcs to intersect each other at E.

(h) Join OE and produce it to M.

Hence, obtuse ∠MOX = 105°.

**Verification:** On measuring by a protractor, we find ∠MOX = 105°.

**(iii) Steps of construction:**

(a) Draw a ray OX with initial point 0.

(h) Taking O as centre and an suitable radius, draw an arc to intersect the ray OX at P and XO produced at D.

(c) Taking P as centre and same radius, draw an arc to intersect the arc drawn in step (b) at A.

(d) Taking A as centre and same radius, draw another arc to intersect the arc drawn in step (b) at B.

(e) Draw the angle bisector of ∠DOB, which intersects the arc drawn in step (b) at C.

(f) Again, draw the angle bisector of ∠COB, which intersects the arc drawn in step (b) at H.

(g) Join OH and produce it to M. Hence, obtuse angle MOX = 135°.

**Verification:** On measuring by the protractor, ∠MOX is of measure 135°.

**5. Construct an equilateral triangle, given its side and justify the construction.**

### Show Answer

**Let the given side of an equilateral triangle is of length 5 cm.**

Sol.Sol.

Steps of construction:

(i) Draw a line segment AB of length 5 cm.

(ii) Draw two angles BAC and ABC, each of measure 60°, on the same side of AB at the points A and B respectively such that their non-common arms intersect each other at C.

Hence, ΔABC is an equilateral triangle.

**Justification:** ∠ABC = ∠BCA = ∠CAB [Each 60°]

⇒ AC = AB = BC.

[Sides opposite to equal angles are equal.]