## CBSE Class 9 Maths Lab Manual – Angle at Centre is Double the Angle Subtended by Same Arc at Any Point on Circumference of Circle

Objective
To verify that the angle subtended by an arc at the centre of circle is double the angle subtended at any point on the remaining part of the circle, experimentally.

Prerequisite Knowledge

1. Basic terms related to circle.
2. Concept of an angle subtended by an arc at the centre and at the circumference of the circle.

Materials Required
Glazed papers, white sheet, pencil, a pair of scissors, gluestick.

Procedure

1. Cut one circle of radius 2.5 cm with centre O from red coloured glazed paper.
2. Cut three more circles from different coloured glazed papers of same radius.
3. Keep all four circles one on the other. Now fold along any part and press them to make a crease. On unfolding, we get chords of same length on each circle.
4. Name the chords, AB in first circle with centre O.
5. Join OA and OB with pencil [fig. (i)].
6. Take two other circles of yellow and green colour and put one on the other and fold it such as to form an angle on the circumference with the same chord AB [fig. (ii) and (iii)].
7. Name these angles as ACB where AB is a chord.
8. Cut angles from fig. (ii) and (iii), ∠ACB from yellow circle and ∠ACB from green circle.
9. Cut the small portion of ∠ACB from both the circles [fig (ii) and (iii)].    10. Paste these two cut outs of fig. (ii) and (iii) on the another (blue) circle [fig. (iv)] at centre O such that their arms lie on the radius OA and OB of circle.

Observation
We observe that two cut outs of angles fully cover ∠AOB in fig.(iv).
∠AOB = ∠ECF + ∠GCH (as ∠ECF = ∠GCH = ∠ACB).
= 2 ∠ACB

Result
Hence we verified that the angle subtended by an arc at the centre of circle is double the angle subtended by the same arc at any point on the remaining part of the circle.

Learning Outcome
Verification of above theorem can be done for arc AB as major arc or semicircular arc. For semicircle, angle on the diameter is of 90°.

Activity Time

1. Prove this theorem by taking different situations on the circle such as chord, a diameter, chord in minor segment, chord on major segment.
2. What do you observe if you take chord as a diameter?

Viva Voce

Question 1.
What is the segment of a circle ?
A chord divides the circle into two parts. Each part is known as segment.

Question 2.
What divides a circle into two equal segments ?
Diameter.

Question 3.
Two diameters of a circle are perpendicular to each other, how many equal sectors will they form in circle ?
4 sectors.

Question 4.
Two chords AB and CD in a circle form ∠AOB and ∠COD at the centre O of the circle. If AB and CD are not equal in length what can you say about the angles ∠AOB and ∠COD ?
∠AOB ≠ ∠COD.

Question 5.
If the angle at the centre is 60°, what will be the angle on the remaining part of circle subtended by an arc ?
30°.

Question 6.
If the angles subtended by the chords of a circle at the centre are equal, then what will be the length of chords ?
Length of the chords will be equal.

Question 7.
What will be the distance of the two equal chords from the centre ?
Equal distance.

Question 8.
If the two chords are equal, then what will be the length of their corresponding arcs ?
Equal length.

Multiple Choice Questions

Question 1.
In a circle, arc BC subtends 40° at the centre, the value of angle subtended by the arc BC at the remaining part of the circle is:
(i) 20°
(ii) 40°
(iii) 80°
(iv) none of these

Question 2.
In a circle, chord AB subtends 32° at circle, the value of angle subtended by it at the centre is:
(i) 16°
(it) 64°
(iii) 30°
(iv) none of these.

Question 3.
Angle subtended by the diameter of the semicircle is:
(i) 90°
(ii) 45°
(iii) 180°
(iv) none of these

Question 4.
Triangle formed by the chord and two radii in a circle is:
(i) right angled triangle
(ii) isosceles triangle
(iii) equilateral triangle
(iv) none of these

Question 5.
In the given figure, O is the centre of the circle, ∠BAC = 40°. Find x: (i) 50°
(ii) 100°
(iii) 80°
(iv) none of these

Question 6.
In the given figure, find x, if ∠ACB = 30°: (i) 60°
(ii) 15°
(iii) 120°
(iv) none of these

Question 7.
The angle formed by a chord is 110° at the centre of the circle, the value of angle formed by it at the remaining part of circle is:
(i) 55°
(ii) 220°
(iii) 100°
(iv) none of these

Question 8.
The angle formed by a chord in minor segment is half of the angle formed by it at centre is:
(i) straight angle
(ii) reflex
(iii) acute angle
(iv) none of these

Question 9.
Find the value of x, in the given figure, if ∠BOC = 42°: (i) 21°
(ii) 84°
(iii) 40°
(iv) none of these

Question 10.
The angle at the centre of the circle is:
(i) 260°
(ii) 180°
(iii) 360°
(iv) none of these