Hint: Draw, identify and differentiate between secant and tangent of a circle in order to prove and apply various theorems related to circles.

Question.1. A figure is shown below. Which of the following is true?

(a) Lines AG and CL are the tangents and line EF is a secant to the circle.
(b) Lines AG and CL are the secants and line EF is a tangent to the circle.
(c) Line AG is a tangent and lines EF and CL are the secants to the circle.
(d) Line AG is a secant and lines EF and CL are the tangents to the circle.

Question.2. A circle passes through point P. How many tangents and secants to the circle are possible that pass through P?

(a) Tangent: 1; Secant: 1
(b) Tangent: Infinite; Secant: 1
(c) Tangent: 1; Secant: Infinite
(d) Tangent: Infinite; Secant: Infinite

Ans.1. (a) Lines AG and CL are the tangents and line EF is a secant to the circle.
Ans.2. (c) Tangent: 1; Secant: Infinite

Hint: Prove and apply theorems related to tangent of a circle in order to determine number of tangents from the given point(s).

Question.3. A circle with centre O is shown below. Which of the following statements is true?

(a) There can be only one line passing through point T such that it is parallel to OT.
(b) There can be only one line passing through point T such that it is perpendicular to OT.
(c) There can be any number of lines passing through point T such that they are parallel to OT.
(d) There can be any number of lines passing through point T such that they are perpendicular to OT.

Question.4. A circle is shown below. Statement I: There is only one line passing through point K which makes an angle of 90° with OK.
Statement II: The shortest distance of a tangent passing through point L from the centre O is equal to the radius of the circle, OL.
Statement III: One tangent can pass through two points K and L of a circle.
Which statement(s) is/are correct?

(a) Statement I and II
(b) Statement I and III
(c) Statement II and III
(d) Statement I, II and III

Ans.3. (b) There can be only one line passing through point T such that it is perpendicular to OT.
Ans.4. (a) Statement I and II

Hint: Prove and apply theorem related to tangent of a circle in order to determine length of the tangent.

Question.5. In the figure shown, RJ and RL are tangents of the circle. What is the measure of \angle JRL?

(a) 90° − 42°
(b) 90° − 84°
(c) 180° − 42°
(d) 180° − 84°

Question.6. In figure is shown, AC and BC are two tangents to the circle O with radius 5 cm. If OC = 13 cm and OD : DC = 3 : 10, then what is the perimeter of triangle ABC?

(a) 24 cm
(b) 30 cm
(c) 32 cm
(d) 34 cm

Ans.5. (d) 180° − 84°
Ans.6. (c) 32 cm