1. **Problem 1:** If arcs APB and CQD of a circle are congruent, find the ratio of AB : CD. Since arcs APB and CQD are congruent, their corresponding chords AB and CD are equal in length. Therefore, the ratio of AB : CD = 1 : 1. 2. **Problem 2:** A and B are points on a circle with center O. C is a point on the circle such that OC bisects \(\angle AOB\). Prove that OC bisects the arc AB. - Given: OC bisects \(\angle AOB\). - Since \(\angle AOC = \angle COB\), the arcs subtended by these angles are equal. - Therefore, arc AC = arc CB. - Hence, OC bisects the arc AB. 3. **Problem 3:** Prove that the angle subtended at the center of a circle is bisected by the radius passing through the midpoint of the arc. - Let M be the midpoint of arc AB. - Radius OM passes through M. - Since M is midpoint, arc AM = arc MB. - Angles subtended by arcs AM and MB at center O are equal. - Therefore, OM bisects \(\angle AOB\). 4. **Problem 4:** In a circle, chords AB and CD intersect at P. If AB = CD, prove that arc AD = arc CB. - Given: AB = CD. - Minor arc AB = minor arc CD (equal chords subtend equal arcs). - Subtract minor arc BD from both sides: \[ \text{arc} AB - \text{arc} BD = \text{arc} CD - \text{arc} BD \] - This simplifies to: \[ \text{arc} AD = \text{arc} CB \] **Multiple Choice Questions with Answers and Explanations:** 1. If P and Q are any two points on a circle, then the line segment PQ is called a - (c) chord of the circle. - Explanation: A chord is a line segment with both endpoints on the circle. 2. If P is a point in the interior of a circle with center O and radius r, then - (d) OP < r. - Explanation: The distance from center O to any interior point P is less than the radius. 3. The circumference of a circle must be - (a) a positive real number. - Explanation: Circumference is the length around the circle, always positive and real. 4. AD is a diameter of a circle and AB is a chord. If AD = 34 cm and AB = 30 cm, then the distance of AB from the center of the circle is - Use Pythagoras theorem in right triangle formed by radius, half chord, and distance from center. - Radius \(r = \frac{AD}{2} = 17\) cm. - Half chord \(= \frac{AB}{2} = 15\) cm. - Distance from center \(= \sqrt{r^2 - (\frac{AB}{2})^2} = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8\) cm. - Answer: (d) 8 cm. 5. If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through points A, B, and C is - Triangle ABC is right angled at B. - Radius is half the hypotenuse AC. - Calculate AC using Pythagoras: \[ AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \] - Radius \(= \frac{AC}{2} = 10\) cm. - Answer: (c) 10 cm.