# NCERT Class 9 Solutions: Constructions (Chapter 11) Exercise 11.1 – Part 1

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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