Class 10 NCERT Maths Chapter 10: Circles
In earlier classes, you learned about circles and related terms like radius, diameter, chord, arc, and tangent. In this chapter, we focus on the properties of tangents to a circle and the theorems related to them.
1. Tangent to a Circle
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact.
Key Property: The tangent to a circle is always perpendicular to the radius drawn at the point of contact.
\[ \text{If } OP \text{ is radius and } PT \text{ is tangent, then } OP \perp PT \]2. Theorems in This Chapter
- Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
3. Tangent Length Formula
If \(O\) is the centre of the circle, \(P\) is an external point, and \(PT\) is the tangent length, then in right-angled triangle \(OPT\):
\[ PT^2 = OP^2 - OT^2 \] where:- \(OP\) = distance from external point to the centre
- \(OT\) = radius of the circle
- \(PT\) = length of the tangent
4. Example Problem
Example: From a point 13 cm away from the centre of a circle, the length of a tangent to the circle is 12 cm. Find the radius of the circle.
Here: \(OP = 13\ \text{cm}\) \(PT = 12\ \text{cm}\) Using \(PT^2 = OP^2 - OT^2\): \[ 12^2 = 13^2 - OT^2 \] \[ 144 = 169 - OT^2 \] \[ OT^2 = 169 - 144 = 25 \] \[ OT = 5 \ \text{cm} \]
5. Applications
- Designing roads and railway tracks where curves meet straight sections.
- Engineering designs involving gears and wheels.
- Optics and reflection problems involving circular mirrors.