# NCERT Class 9 Solutions: Constructions (Chapter 11) Exercise 11.1 – Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Construction of point angle at

Q-1 Construct an angle of at the initial point of a given of ray and justify the construction.

Solution:

To construct an angle at point A

Steps of construction:

- Draw a ray AS.
- With its initial point A as centre and any radius, draw an arc BCD, cutting AS at B.
- With same radius and center at B, cut the arc intersecting the circle with center A at C
- With centre C and same radius, draw an arc, cutting the circle with center A at D.
- With D and C as centres, and any convenient radius () draw two arcs intersecting at P
- Join AP. Then

Justification:

- AB and AC are the radius and AB = BC by construction, therefore,
- Therefore is an equilateral triangle. So,
- Similarly, , therefore is also an equilateral triangle. So
- AP bisects , so
- Now

Q-2 Construct an angle of at the initial point of a given ray and justify the construction.

Solution:

Steps of Construction:

- Draw a ray OA
- With O as centre and any suitable radius draw an arc cutting OA at B.
- With B as centre and same radius cut the previously drawn arc at C and then with C as centre and same radius cut the arc in step 1 at D.
- With C as centre and radius more than half CD draw an arc.
- With D as centre and same radius draw another arc to cut the previous arc at E
- Join OE. Then (we proved this above) . Let OE cut the circle with center O at F

Now we will draw the bisector OG of Then

- With B as center and radius more than half of BF draws an arc.
- With F as center and radius more than half of BF draws another arc intersecting previous arc at G.
- Now OG is the bisector of angle . Since is the bisector of . So,

Justification:

- We already proved that constructed this way would be .
- Now, in and ,
- OG is common
- (by construction)
- (radius of the same circle at O)

- Therefore, by SSS
- Therefore,
- Also,
- Therefore,

Q-3 Construct the angle of the following measurements:

1.

2.

3.

Solution:

**i) Steps of Construction**:

- Draw a ray AQ.
- With its initial point A as centre and any radius, draw an arc, cutting AQ at P.
- With centre P and same radius. Draw an arc, cutting the arc of step 2 in D
- With P and D as centres, and any convenient radius () , draw two arcs intersecting at B.
- Join OB. Then

**ii) Step of construction**:

- Draw an angle
- Draw the bisector OC of , then
- Bisect , such that
- Thus,

**iii) Steps of Construction**:

- Construct an
- Bisect so that
- Bisect , so that