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

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

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

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

Procedure
Case I.

1. Draw a circle of any radius with centre O on a glazed paper. Cut it and paste it on white paper.
2. Fold the circle along the line passing through the centre O to get a diameter AB.
   
3. Take any point P on circumference of the circle.
4. Join AP and BP by paper folding to get ∠APB.
5. Make two replicas of ∠APB with the help of tracing paper such that ∠A₁P₁B₁ and ∠A₂P₂B₂.
6. Place two ∆A₁P₁B₁ and ∆A₂P₂B₂ such that ∠P₁ and ∠P₂ coincide each other [fig.(ii)].
   

We notice ∠A₁P₁B₁ and ∠A₂P₂B₂ form a linear pair.
∴ ∠A₁P₁B₁ + ∠A₂P₂B₂ = 180° (linear pair).
2∠APB = 180° (∠A₁P₁B₁ and ∠A₂P₂B₂ are replicas of ∠APB)
∴ ∠APB = 90°

Case II. For major segment:

1. Cut a circle of any radius using glazed paper with centre O and paste it on a white paper.
2. Make a chord AB by paper folding.
3. Take a point Q on the major segment. Join QA and QB by paper folding.
4. Draw and cut replica of ∠AQB.
5. 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:

1. Cut a circle of any radius using glazed paper with centre O. Paste it on white paper.
2. Make a chord AB by paper folding.
3. Take any point M on the minor segment. Join MA and MB by paper folding to get ∠AMB.
4. Draw and cut replica of ∠AMB with the help of tracing paper.
5. 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:

1. Along the diameter and measure different angles formed on the diameter by paper folding method.
2. 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 cm²
(ii) 6 cm²
(iii) 2 cm²
(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

1. (i)
2. (i)
3. (i)
4. (i)
5. (iii)
6. (i)
7. (i)
8. (ii)
9. (i)
10. (iii)

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