Tangents and Secants to a Circle

Tangents and Secants to a Circle

• The line PQ and the circle have no common point. In this case PQ is a nonintersecting line with respect to the circle.
• The line PQ intersects the circle at two points A and B. It forms a chord AB on the circle with two common points. In this case the line PQ is a secant of the circle.
• There is only one point A, common to the line PQ and the circle. This line is called a tangent to the circle.

The common point of the tangent and the circle is called the point of contact and the tangent is said to touch the circle at the common point.

• We can draw infinite tangents to the circle.
• We can draw two tangents to the circle from a point away from it.
• The tangent at any point of a circle is perpendicular to the radius through the point of contact.

EXERCISE-9.I

1. Fill in the blanks
   • i) A tangent to a circle intersects it in ….one………… point (s).
   • ii) A line intersecting a circle in two points is called a………secant……
   • iii) The number of tangents drawn at the end points of the diameter is……. two…….
   • iv) The common point of a tangent to a circle and the circle is called ……. point of contact…..
   • v) We can draw…. infinite….. tangents to a given circle.
2. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 13 cm. Find length of PQ.

𝑃𝑄² + 𝑂𝑃² = 𝑂𝑄²

𝑃𝑄² + 5² = 13²

𝑃𝑄² + 25 = 169

𝑃𝑄² = 169 − 25 = 144 = 12²

∴ 𝑃𝑄 = 12 𝑐𝑚

3. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

∠𝑂𝐴𝑃 = 90⁰ 𝑎𝑛d ∠𝑂𝐵𝑆 = 90⁰

⇒ ∠𝐵𝐴𝑃 = 90⁰ 𝑎𝑛𝑑 ∠𝐴𝐵𝑆 = 90⁰

⇒ ∠𝐵𝐴𝑃 = ∠𝐴𝐵𝑆

⇒ 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑒 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙

⇒ 𝑃𝑄 ∥ 𝑅𝑆

Prob1:Prove The centre of a circle lies on the bisector of the angle between two tangents drawn from a point outside it.