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:

Ray as with the Initial Point A

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:

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

Solution:

Ray OA, O is the Centre of the 45° Angle

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:

Solution:

i) Steps of Construction:

∠QAB=30°

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:

POB=90°

iii) Steps of Construction:

Ray AB at ∠CAB=15°