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 ∠A1P1B1 and ∠A2P2B2.
- 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:
- 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 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
- (i)
- (i)
- (i)
- (i)
- (iii)
- (i)
- (i)
- (ii)
- (i)
- (iii)
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