Welcome to your ultimate and comprehensive study guide for Chapter 10: Tangent to a Circle. Circle geometry is an essential component of mathematical reasoning. This chapter focuses on secants, tangents, points of contact, perpendicularity between tangents and radial segments, lengths of tangents drawn from external points, and properties of touching circles.
1. Definitions: Secant and Tangent
- Secant: A straight line that cuts the circumference of a circle in two distinct points.
- Tangent: A straight line that touches the circumference of a circle at a single point only, known as the point of contact or point of tangency.
2. Key Theorems on Tangents
- Theorem 1: If a line is drawn perpendicular to a radial segment of a circle at its outer end point, it is a tangent to the circle at that point.
- Theorem 2: The tangent to a circle and the radial segment joining the point of contact and the centre are perpendicular to each other.
- Theorem 3: The two tangents drawn to a circle from a point outside it are equal in length.
- Theorem 4: If two circles touch each other externally or internally, the distance between their centres is respectively equal to the sum or difference of their radii.
3. Step-by-Step Example: Equal Tangents from an External Point
Example: Understanding properties of two tangents drawn from an external point P.
Step-by-Step Solution:
- Step 1: Identify given elements: Two tangents PA and PB are drawn from an external point P to a circle with centre O.
- Step 2: Construct triangles: Join O with A, B, and P to form right-angled triangles OAP and OBP.
- Step 3: Apply Hypotenuse-Side (H.S) Postulate: Since OA = OB (radii) and OP is a common hypotenuse, triangle OAP is congruent to triangle OBP.
- Step 4: Conclude lengths: Therefore, PA = PB, proving that tangents from an external point are equal in length.
Essential Conceptual Review Questions
Q1: What is the angle between a tangent and the radial segment at the point of contact?
Answer: The angle is exactly 90 degrees, meaning the tangent and the radial segment are strictly perpendicular to each other.
Q2: How is the distance between the centres of two externally touching circles calculated?
Answer: It is equal to the sum of the radii of the two individual circles (distance = r1 + r2).