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

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Textbook | NCERT |

Board | CBSE |

Category | NCERT Solutions |

Class | Class 9 |

Subject | Maths |

Chapter | Chapter 12 |

Exercise | Class 9 Chapter 12 Constructions Exercise 12.1 |

Number of Questions Solved | 6 |

## 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**

- Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
- Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
- Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
- Draw the ray OE passing through C.

Then, ∠EOA = 60° - Draw the ray of passing through D. Then, ∠FOE = 60°.
- Next, taking Cand Das centres and with the radius more than 1/2 CD, draw arcs to intersect each other, say at G.
- 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**

- Taking O as centre and some radius, draw an arc of a circle which intersects OA, say at a point B.
- Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
- Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
- Draw the ray OE passing through C. Then, ∠EOA = 60°.
- Draw the ray OF passing through D. Then, ∠FOE = 60°.
- Next, taking C and D as centres and with radius more than 1/2 CD, draw arcs to intersect each other, say at G.
- 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° - Now taking O as centre and any radius, draw an arc to intersect the rays OA and OG, say at Hand /, respectively.
- Next, taking H and las centres and with the radius more than 1/2 Hl, draw

arcs to intersect each other, say at J. - 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**

- Taking O as centre and some radius, draw an arc of a circle which intersects OA, say at a point B.
- Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C.
- Draw the ray OE passing through C. Then, ∠EOA = 60°.
- Taking B and C as centres and with the radius more than 1/2 BC, draw arcs to intersect each other, say at D.
- 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**

- Taking O as centre and some radius, draw an arc of a circle which, intersects OA, say at a point B.
- Taking C as centre and with the same radius as before, drawn an arc intersecting the arc drawn in step 1, say at D.
- Draw the ray OE passing through C. Then, ∠EOA = 60°.
- Draw the ray OF passing through D. Then, ∠FOE = 60°.
- Next, taking C and D as centres and with radius more than 1/2CD, draw arcs to intersect each other, say at G.
- Draw the ray OG. This ray OG is the bisector of the ∠FOE,
- Now, taking O as centre and any radius, draw an arc to intersect the rays OA and OG, say at H and l, respectively.
- Next, taking H and l as centres and with the radius more than 1/2Hl, draw arcs to intersect each other, say at J.
- Draw the ray OJ. This ray OJ is the bisector of the ∠GOA

i. e., ∠GOJ = ∠AOJ = 1/2 ∠GOA

= 1/2 (90°) = 45 ° - Now, taking O as centre and any radius, drawn an arc to intersect the rays OA and OJ, say at K and L, respectively.
- Next, taking K and L as centres and with the radius more than 1/2KL, draw arcs to intersect each other, say at H.
- 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**

- Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
- Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point C..
- Draw the ray OE passing through C. Then, ∠EOA = 60°.
- Now, taking 6 and Cas centres and with the radius more than 1/2 BC, draw arcs to intersect each other, say at D.
- 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° - 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.
- Draw the ray OG. This ray OG is the bisector of the ∠AOD,

i. e., ∠DOG = ∠AOG = 1/2 ∠AOD = 1/2 (30°) = 15°

- Draw the ray OG. This ray OG is the bisector of the ∠AOD,

**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**

- Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
- Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.
- Join the ray OE passing through C. Then, ∠EOA = 60°.
- Draw the ray of passing through D. Then, ∠FOE = 60°.
- Next, taking C and D as centres and with the radius more than 1/2CD, draw arcs to intersect each other, say at G.
- 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° - 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**

- Taking O as centre and some radius, draw an arc of a circle which intersects OA say at a point B.
- 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.
- Draw the ray Of passing through C. Then, ∠EOA = 60°.
- Draw the ray OF passing through D. Then, ∠FOE = 60°.
- Next, taking Cand Das centres and with the radius more than 1/2 CD, draw arcs to intersect each other, say at G.
- Draw the ray OG intersecting the arc drawn in step 1 at H. This ray OG is the bisector of the ∠FOE, i.e.,
- 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.
- 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**

- Produce AO to A’ to form ray OA’.
- 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’.
- Taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at a point C.
- Draw the ray OE passing through C, then ∠EOA = 60°.
- Draw the ray OF passing through D, then ∠FOE = 60°.
- 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.
- Draw the ray OGintersecting the arc drawn in step 1 at H. This ray OG is the bisector of the ∠FOE i,e.,
- 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.
- 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**

- Take a ray AX with initial point A From AX, cut off AB = 4 cm.
- Taking A as centre and radius (= 4 cm), draw an arc of a circle, which intersects AX, say at a point B.
- Draw the ray AE passing through C.
- Next, taking B as centre and radius (= 4 cm), draw an arc of a circle, which intersects AX, say at a point A
- 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

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