**NCERT Class 9 Maths Lab Manual – Verify that the Angle Subtended by an Arc of a Circle**

**OBJECTIVE**

To verify that the angle subtended by an arc of a circle at the centre is double the angle subtended by it at an point on the remaining part of the circle.

**Materials Required**

- Coloured drawing sheets
- Cardboard
- Geometry box
- White paper
- Adhesive
- Transparent sheet
- Cutter/Scissors

**Prerequisite Knowledge**

- All the basic knowledge related to the circle.
- Angle subtended by an arc.

**Theory**

- The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle. The fixed point is called the centre of the circle, the line segment joining the centre and any point on the circle is called radius of circle.
- A line segment joining two points on the circle is called a chord of the circle.
- A chord which passes through the centre of the circle is called a diameter of the circle.
- The length of the complete circle is called its circumference.
- A piece of a circumference of circle between two points is called an arc.

- Angle Subtended by an Arc of a Circle

Let we have a circle with centre at O and AB be its arc. Here, ∠AOB is the angle subtended by arc AB (\(\bar { AB }\) ) at the centre of the circle.

Also, ∠APS is the angle subtended by arc AB (\(\bar { AB }\)) at a point P on the remaining part of the

circle. - Important Points about Angle Subtended by an Arc
- The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal.
- Angle in a semi-circle is a right angle.
- If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).

**Procedure**

- Take a rectangular piece of cardboard of suitable size and by using adhesive, paste a white paper on it.
- Cut out a circle of suitable radius with centre O from drawing sheet and paste it on the cardboard.
- Take a pair of points O and R on the circle to obtain the arc QR. (see Fig. 23.3)

- To obtain the angle subtended by arc QR at centre O, join the points O and R to the centre O. (see Fig. 23.3)
- Taking a point P on the remaining part of circle, join it to Q and R to get ∠QPR subtended by arc QR on point P on the remaining part of circle, (see Fig. 23.3)
- Mark ∠QPR and ∠QOR.
- Make a cut out of ∠QOR and a pair of cut outs of ∠QPR using transparent sheet, (see Fig. 23.4)

- Now, place the pair of cut outs of ∠QPR on the cut out of ∠QOR, adjacent to each other, (see Fig. 23.5)

**Demonstration**

Flere, ∠QOR = 2 ∠QPR

We find that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of circle.

**Observation**

By actual measurement, ∠QOR = ………… ,

∠QPR = ………… ,

Therefore, ∠QOR = 2 ………… .

**Result**

We find that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

**Application**

The property is used in proving many other important results such as opposite angles of a cyclic quadrilateral are always supplementary angles and angles in the same segment of a circle are always equal.

**Viva Voce**

**Question 1:**

Define a circle.

**Answer:**

A circle is the collection of all points in a plane which are equidistant from a fixed point within the same plane.

**Question 2:**

What is chord?

**Answer:**

A line segment which is formed by joining any two points on the circumference of circle.

**Question 3:**

How many circles can pass through three non-collinear points?

**Answer:**

There is only one such circle.

**Question 4:**

What is the diameter?

**Answer:**

The chord which passes through the centre of circle is known as the diameter of circle.

**Question 5:**

Which is the longest chord of circle?

**Answer:**

Diameter

**Question 6:**

If the angle subtended by an arc at centre is 110°, then what will be the angle on the remaining part of circle subtended by same arc?

**Answer:**

55°

**Question 7:**

If two chords are equal, then what will be the lengths of their corresponding arcs?

**Answer:**

For equal chords, the lengths of their corresponding arcs are always equal.

**Question 8:**

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

**Answer:**

Both chords will be of equal length.

**Question 9:**

What will be the distance of two equal chords from the centre?

**Answer:**

Both chords are at equal distance from the centre.

**Suggested Activity**

To verify the converse of this theorem experimentally.

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