# NCERT Solutions For Class 9 Maths Chapter 12 Constructions Ex 12.1

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## NCERT Solutions for Class 9 Maths Chapter 12 Constructions Ex 12.1

NCERT TEXTBOOK EXERCISES

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

Solution:

Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
3. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
4. Draw the ray OE passing through C.
Then, ∠EOA = 60°
5. Draw the ray of passing through D. Then, ∠FOE = 60°.
6. Next, taking Cand Das centres and with the radius more than 1/2 CD, draw arcs to intersect each other, say at G.
7. Draw the ray OG. This ray OG is the bisector of the ∠FOE i.e.,
∠FOG = ∠EOG = 1/2 ∠FOE = 1/2 (60°) = 30°
Thus, ∠GOA = ∠GOE + ∠EOA
= 30° + 60° = 90°

Justification
(i) Join BC.
Then, OC=OB = BC (By construction)
∴ ∆COB is an equilateral triangle.
∴ ∠COB = 60°
∴ ∠EOA = 60°

(ii) Join CD.
Then, OD=OC=CD (By construction)
∴ ∆DOC is an equilateral triangle.
∴ ∠DOC = 60°
∴ ∠FOE = 60°

(iii) Join CG and DG.
In ∆ODG and ∆OCG,
OD = OC (Radii of the same arc)
DG = CG (Arcs of equal radii)
OG = OG (Common)
∴ ∆ ODG ≅ ∆OCG (SSS rule)
∴ ∠DOG = ∠COG (CPCT)
∴ ∠FOG = ∠EOG = 1/2 ∠FOE
= 1/2 (60°) = 30°
Thus, ∠GOA = ∠GOE + ∠EOA = 30° + 60° = 90°

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

Solution:

Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which intersects OA, say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
3. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
4. Draw the ray OE passing through C. Then, ∠EOA = 60°.
5. Draw the ray OF passing through D. Then, ∠FOE = 60°.
6. Next, taking C and D as centres and with radius more than 1/2 CD, draw arcs to intersect each other, say at G.
7. Draw the ray OG. This ray OG is the bisector of the ∠FOE,
i.e., ∠FOG = ∠EOG = 1/2 ∠FOE = 1/2 (60°) = 30°.
thus , ∠GOA = ∠GOE + ∠EOA
= 30° + 60° = 90°
8. Now taking O as centre and any radius, draw an arc to intersect the rays OA and OG, say at Hand /, respectively.
9. Next, taking H and las centres and with the radius more than 1/2 Hl, draw
arcs to intersect each other, say at J.
10. Draw the ray OJ. This ray OJ is the required bisector of the ∠GOA.
∠GOJ = ∠AOJ = 1/2 ∠GOA
= 1/2 (90°) = 45°

Justification
(i) Join BC. (By construction)
Then, OC = OB = BC
∴ ∆COB is an equilateral triangle.
∴ ∠COB = 60°
∴ ∠EOA = 60°

Question 3. Construct the angles of the following measurements

(i) 30°
(ii) 22 12
(iii) 15°

Solution:

(i) Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which intersects OA, say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
3. Draw the ray OE passing through C. Then, ∠EOA = 60°.
4. Taking B and C as centres and with the radius more than 1/2 BC, draw arcs to intersect each other, say at D.
5. Draw the ray OD, this ray OD is the bisector of the ∠EOA, i.e.,
∠EOD = ∠AOD = 1/2 ∠EOA = 1/2 (60°) = 30°

(ii) Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which, intersects OA, say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
3. Taking C as centre and with the same radius as before, drawn an arc intersecting the arc drawn in step 1, say at D.
4. Draw the ray OE passing through C. Then, ∠EOA = 60°.
5. Draw the ray OF passing through D. Then, ∠FOE = 60°.
6. Next, taking C and D as centres and with radius more than 1/2CD, draw arcs to intersect each other, say at G.
7. Draw the ray OG. This ray OG is the bisector of the ∠FOE,
8. Now, taking O as centre and any radius, draw an arc to intersect the rays OA and OG, say at H and l, respectively.
9. Next, taking H and l as centres and with the radius more than 1/2Hl, draw arcs to intersect each other, say at J.
10. Draw the ray OJ. This ray OJ is the bisector of the ∠GOA
i. e., ∠GOJ = ∠AOJ = 1/2 ∠GOA
= 1/2 (90°) = 45 °
11. Now, taking O as centre and any radius, drawn an arc to intersect the rays OA and OJ, say at K and L, respectively.
12. Next, taking K and L as centres and with the radius more than 1/2KL, draw arcs to intersect each other, say at H.
13. Draw the ray OM. This ray OM is the bisector of the ∠AOJ, i.e., ∠JOM = ∠AOM
= 1/2 ∠AOJ = 1/2 (45°) = 22 1/2 °

(iii) Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C..
3. Draw the ray OE passing through C. Then, ∠EOA = 60°.
4. Now, taking 6 and Cas centres and with the radius more than 1/2 BC, draw arcs to intersect each other, say at D.
5. Draw the ray OD intersecting the arc drawn in step 1 at F. This ray OD is the bisector of the ∠EOA, i.e.,
∠EOD = ∠AOD = 1/2 ∠EOA = 1/2 (60°) = 30°
6. Now, taking B and F as centres and with the radius more than 1/2 BF, draw arcs to intersect each other, say at G.
1. Draw the ray OG. This ray OG is the bisector of the ∠AOD,
i. e., ∠DOG = ∠AOG = 1/2 ∠AOD = 1/2 (30°) = 15°

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

(i) 75°
(ii) 105°
(iii) 135°

Solution:

(i) Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
3. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
4. Join the ray OE passing through C. Then, ∠EOA = 60°.
5. Draw the ray of passing through D. Then, ∠FOE = 60°.
6. Next, taking C and D as centres and with the radius more than 1/2CD, draw arcs to intersect each other, say at G.
7. Draw the ray OG intersecting the arc of step 1 at H. This ray OG is the bisector of the ∠FOE, i.e., ∠FOG = ∠EOG
= 1/2 ∠FOE = 1/2(60°) = 30°
8. Next, taking Cand H as centres and with the radius more than 1/2CH, draw
arcs to intersect each other, say at l.
Draw the ray OI. This ray OI is the bisector of the ∠GOE,
i. e., ∠GOI = ∠EOI = 1/2 ∠GOE = 1/2 (30°) = 15°
Thus, ∠IOA = ∠IOE + ∠EOA
=15°+ 60° = 75°
On measuring the ∠IOA by protractor, we find that ∠IOA = 15°
Thus, the construction is verified.

(ii) Steps of construction

1. Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
2. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
3. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at a point D.
4. Draw the ray Of passing through C. Then, ∠EOA = 60°.
5. Draw the ray OF passing through D. Then, ∠FOE = 60°.
6. Next, taking Cand Das centres and with the radius more than 1/2 CD, draw arcs to intersect each other, say at G.
7. Draw the ray OG intersecting the arc drawn in step 1 at H. This ray OG is the bisector of the ∠FOE, i.e.,
8. Next, taking H and D as centres and with the radius more than 1/2 HD, draw arcs to intersect each other, say at l.
9. Draw the ray Ol. This ray Ol is the bisector of the ∠FOG, i.e.,
Thus, ∠lOA = ∠IOG + ∠GOA = 15° + 90° = 105°. On measuring the ∠lOA by protractor, we find that ∠FOA = 105°.
Thus, the construction is verified.

(iii) Steps of construction

1. Produce AO to A’ to form ray OA’.
2. Taking O as centre and some radius, draw an arc of a circle which intersects OA at a point B and OA’ at a point B’.
3. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at a point C.
4. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
5. Draw the ray OE passing through C, then ∠EOA = 60°.
6. Draw the ray OF passing through D, then ∠FOE = 60°.
7. Next, taking C and D as centres and with the radius more than 1/2 CD, draw arcs to intersect each other, say at G.
8. Draw the ray OGintersecting the arc drawn in step 1 at H. This ray OG is the bisector of the ∠FOE i,e.,
9. Next, taking B’ and H as centres and with the radius more than 1/2 B’H, drawn arcs to intersect each other, say at l.
10. Draw the ray Ol. This ray Ol is the bisector of the ∠B’OG i.e.,
On measuring the ∠IOA by protractor, we find that ∠lOA = 135°.
Thus, the construction is verified.

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

Solution:

Steps of construction

1. Take a ray AX with initial point A From AX, cut off AB = 4 cm.
2. Taking A as centre and radius (= 4 cm), draw an arc of a circle, which intersects AX, say at a point B.
3. Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
4. Draw the ray AE passing through C.
5. Next, taking B as centre and radius (= 4 cm), draw an arc of a circle, which intersects AX, say at a point A
6. Taking A as centre and with the same radius as in step 5, draw an arc intersecting the previously drawn arc, say at a point C.
Draw the ray BF passing through C.
Then, ∆ ABC is the required triangle with gives side 4 cm.

Justification
AB = BC (By construction)
AB = AC (By construction)
∴ AB = BC = CA
∴ ∆ ABC is an equilateral triangle.
∴ The construction is justified.

## NCERT Solutions for Class 9 Maths Chapter 12 Constructions Exercise 12.1 PDF

For NCERT Solutions for Class 9 Maths Chapter 12 Constructions Ex 12.1, you may click on the link below and get your NCERT Solutions for Class 9 Maths Chapter 12 Circles Exercise pdf file.