**CBSE Class 9 Maths Lab Manual – Angle in a Semicircle, Major Segment, Minor Segment**

**Objective**

To verify that angle in a semicircle is a right angle, angle in a major segment is acute, angle in a minor segment is obtuse by paper folding.

**Prerequisite Knowledge**

- Concept of semicircle, major segment and minor segment
- Concept of right angle, acute angle and obtuse angle.

**Materials Required**

White sheet, glazed papers, compass, pencil, tracing paper

**Procedure**

**Case I.**

- Draw a circle of any radius with centre O on a glazed paper. Cut it and paste it on white paper.
- Fold the circle along the line passing through the centre O to get a diameter AB.

- Take any point P on circumference of the circle.
- Join AP and BP by paper folding to get ∠APB.
- Make two replicas of ∠APB with the help of tracing paper such that ∠A
_{1}P_{1}B_{1}and ∠A_{2}P_{2}B_{2}. - Place two ∆A
_{1}P_{1}B_{1}and ∆A_{2}P_{2}B_{2}such that ∠P_{1}and ∠P_{2}coincide each other [fig.(ii)].

We notice ∠A_{1}P_{1}B_{1} and ∠A_{2}P_{2}B_{2} form a linear pair.

∴ ∠A_{1}P_{1}B_{1} + ∠A_{2}P_{2}B_{2} = 180° (linear pair).

2∠APB = 180° (∠A_{1}P_{1}B_{1} and ∠A_{2}P_{2}B_{2} are replicas of ∠APB)

∴ ∠APB = 90°

**Case II. For major segment:**

- Cut a circle of any radius using glazed paper with centre O and paste it on a white paper.
- Make a chord AB by paper folding.
- Take a point Q on the major segment. Join QA and QB by paper folding.
- Draw and cut replica of ∠AQB.
- Place the replica of ∠AQB on the newly drawn, right angled ∆DEF such that side BQ falls on DE.

∴ ∠AQB < ∠DEF = 90°

∴ ∠AQB is acute.

**Case III. For minor segment:**

- Cut a circle of any radius using glazed paper with centre O. Paste it on white paper.
- Make a chord AB by paper folding.
- Take any point M on the minor segment. Join MA and MB by paper folding to get ∠AMB.
- Draw and cut replica of ∠AMB with the help of tracing paper.
- Place the replica of ∠AMB on the base of newly drawn right angled triangle ∆DEF, such that base MB coincides with EF and point M coincides with E.

Here, ∠AMB > ∠DEF = 90°

∴ ∠AMB is an obtuse angle.

**Observation**

We observe that

In Case I, AOB is a diameter and ∠APB is of 90°.

In Case II, AQB is a major segment and ∠AQB is an acute angle.

In Case III, AMB is a minor segment and ∠AMB is an obtuse angle.

**Result**

By paper folding method, we verified that angle in a semicircle is right angle. In any circle, angle in minor segment is obtuse angle, angle in major segment is an acute angle.

**Learning Outcome**

In any circle, any angle in a minor segment is always obtuse, any angle in a major segment is always acute, angle in semicircle is always a right angle.

**Activity Time**

Divide the circle into two parts:

- Along the diameter and measure different angles formed on the diameter by paper folding method.
- Along any chord (other than diameter) and measure the different angles formed by paper folding on two different segments.

**Viva Voce**

**Question 1.**

What is a semicircle ?

**Answer:**

Half of the circle.

**Question 2.**

If AB is any chord of a circle, what will be the sum of the angle in minor segment and major segment ?

**Answer:**

180°.

**Question 3.**

A circle has finite number of equal chords. Is this statement true.

**Answer:**

False.

**Question 4.**

What type of angle is obtained in minor segment ?

**Answer:**

Obtuse angle.

**Question 5.**

What type of angle is obtained in major segment?

**Answer:**

Acute angle.

**Question 6.**

What is the longest chord in a circle ?

**Answer:**

Diameter.

**Question 7.**

If the angle subtended by a chord in major segment is acute angle what will be angle at the centre ?

**Answer:**

Angle will be less than 180°.

**Question 8.**

If angle subtended by an arc is 70° in alternate segment, in which segment angle will lie ?

**Answer:**

It will lie in a major segment.

**Question 9.**

If the angle subtended by an arc is 110° in alternate segment, in which segment angle will lie ?

**Answer:**

It will lie in a minor segment.

**Multiple Choice Questions**

**Question 1.**

If two equal chords AB and CD intersect each other inside the circle at E. Then, arc AD and arc CD are:

(i) congruent

(ii) not-congruent

(iii) \(\widehat { AD }\) = 2\(\widehat { CB }\)

(iv) none of these

**Question 2.**

In any circle, measure of angle in minor segment is 110°, what is value of its opposite angle on the circle:

(i) 70°

(ii) 60°

(iii) 55°

(iv) none of these

**Question 3.**

In any circle, for a chord, the sum of opposite angles subtended by it in major segment and minor segment is:

(i) 180°

(ii) 360°

(iii) 90°

(iv) none of these

**Question 4.**

AB is a chord of a circle with centre O. OP = 3 cm if radius is 5 cm, OP ⊥ AB, then length of the chord AB will be:

(i) 8 cm

(ii) 4 cm

(iii) 16 cm

(iv) none of these

**Question 5.**

On a semicircle with AB as diameter, a point C is taken so that ∠CAB = 30°. Find ∠ACB:

(a) 30°

(ii) 60°

(iii) 90°

(iv) none of these

**Question 6.**

On a semicircle with PQ as diameter, a point B is taken so that ∠BPQ = 60°. Find ∠BQP:

(i) 30°

(ii) 90°

(iii) 120°

(iv) none of these

**Question 7.**

PQ and RS are two parallel chords of a circle and lines RP and SQ intersect each other at O outside the circle. Then, OP and OQ are:

(i) equal

(ii) not equal

(iii) OP = 3OQ

(iv) none of these

**Question 8.**

Find the area of a right angled triangle, if the radius of its circumcircle is 3 cm and altitude drawn to the hypotenuse is 2 cm:

(i) 3 cm^{2}

(ii) 6 cm^{2}

(iii) 2 cm^{2}

(iv) none of these

**Question 9.**

If the angle in major segment is acute, then angle opposite to it will be:

(i) Obtuse

(ii) right angle

(iii) acute

(iv) none of these

**Question 10.**

If the angle in minor segment is obtuse then other angle on the same segment will be:

(i) right angle

(ii) acute

(iii) obutse

(iv) none of these

**Answers**

- (i)
- (i)
- (i)
- (i)
- (iii)
- (i)
- (i)
- (ii)
- (i)
- (iii)

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