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

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Construction of point angle at

Construction of point angle at 120 degree

Construction of Point Angle at 120 Degree

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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:

  • 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:

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:

1.

2.

3.

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°

  • Draw an angle

  • Draw the bisector OC of , then

  • Bisect , such that

  • Thus,

iii) Steps of Construction:

Ray AB at ∠CAB=15°

  • Construct an

  • Bisect so that

  • Bisect ,so that