Q1. In a circle with centre O, the central angle AOB is \( 60^{\circ} \). If the radius of the circle is \( 12 \mathrm{cm} \), what is the length of the chord AB?

Answer:

Given: Radius (r) = 12 cm, central angle = 60°.

Triangle AOB is an isosceles triangle with OA = OB = 12 cm and \( \angle AOB = 60° \).

Since OA = OB, the base angles \( \angle OAB \) and \( \angle OBA \) are equal.

\( \angle OAB = \angle OBA = \frac{180° - 60°}{2} = 60° \)

Therefore, triangle AOB is an equilateral triangle (all angles are 60°).

Hence, AB = OA = OB = 12 cm.

The length of chord AB is 12 cm.

Q2. Let A and B be two points on a circle with centre O. (i) Are there points X, Y on the circle, on the same side of AB, such that \( \angle \mathrm{AXB} \) is different from \( \angle \mathrm{AYB} \)? (ii) Is it true that if \( \angle \mathrm{AXB} = \angle \mathrm{AYB} \), then X and Y lie on the same side of the circle? (iii) If \( \angle \mathrm{AXB} = \angle \mathrm{AYB} \), and X and Y do not lie on the circle, does the circle through A, B and X also pass through Y?

Answer:

(i) No. For any two points X and Y on the same side of chord AB, the angles \( \angle AXB \) and \( \angle AYB \) are equal. This is because they are angles in the same segment of the circle (Theorem 9).

(ii) No. If \( \angle AXB = \angle AYB \), X and Y could be on the same side or on opposite sides of AB. If they are on opposite sides, the angles they subtend are supplementary (they add up to 180°).

(iii) Yes. If \( \angle AXB = \angle AYB \), and X and Y are on the same side of AB, then by Theorem 10, A, B, X, and Y are concyclic. If X and Y are on opposite sides, then they are also concyclic but lie on different arcs of the same circle.

Q3. Find \( x \) in Fig.5.26. [IMAGE NEEDED: Figure 5.26 from the textbook showing a circle with angle x to be found]

Answer:

I cannot see the figure, but the problem likely involves using the property that angles in the same segment are equal, or that the angle subtended by an arc at the centre is twice the angle subtended at the circumference.

For example, if the figure shows a central angle of 100° subtending an arc, then the angle at the circumference (x) would be 50°.

If the figure shows two angles in the same segment, they would be equal.

Please look at the figure and apply the relevant circle theorem:

• Angle at centre is twice the angle at circumference (when subtended by the same arc).
• Angles in the same segment are equal.