Tangent–Radius Theorem (Prove) MCQs Quiz | Class 10

Understanding the Tangent–Radius Theorem

The Tangent–Radius Theorem is a foundational concept in Euclidean geometry, particularly when studying circles. It establishes a critical relationship between a circle’s tangent and its radius at the point of contact.

Key Concepts and the Theorem Statement

A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact. A radius is a line segment connecting the center of the circle to any point on its circumference.

Tangent–Radius Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

This means that if a line AB is tangent to a circle with center O at point P, then the radius OP is perpendicular to the line AB (i.e., angle OPA = 90 degrees).

Proof of the Tangent–Radius Theorem

1. Consider a circle with center O and a tangent XY at point P on the circle.
2. We need to prove that OP is perpendicular to XY.
3. Take any point Q, other than P, on the tangent XY.
4. Join OQ.
5. Since Q is a point on the tangent XY (other than P), Q must lie outside the circle. (If Q were inside, XY would be a secant, not a tangent.)
6. Therefore, OQ must be longer than the radius OP (as OP is the distance from O to a point on the circle, and OQ is the distance from O to a point outside the circle). So, OQ is greater than OP.
7. This holds true for every point on the line XY, except for the point P itself.
8. Thus, OP is the shortest distance from the center O to any point on the line XY.
9. We know that the shortest distance from a point to a line is the perpendicular distance.
10. Hence, OP is perpendicular to XY.
11. This completes the proof.

Corollaries and Applications

• The converse of the theorem is also true: A line drawn through the end point of a radius and perpendicular to it is tangent to the circle.
• From an external point, two tangents can be drawn to a circle. These two tangents are equal in length.
• The line segment joining the center to the external point bisects the angle between the two tangents.
• The line segment joining the center to the external point bisects the angle between the radii to the points of contact.

Properties of Tangents from an External Point

If two tangents PA and PB are drawn from an external point P to a circle with center O:

Quick Revision Checklist

• A tangent touches a circle at only one point.
• The radius at the point of contact is always perpendicular to the tangent.
• The perpendicular from the center to a tangent passes through the point of contact.
• Two tangents from an external point to a circle are equal in length.
• The line joining the center to the external point bisects the angle between the tangents and the angle between the radii.

Practice Questions

1. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q so that OQ = 12 cm. Find the length of PQ.
2. In two concentric circles, the radius of the larger circle is 5 cm and that of the smaller circle is 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
3. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the center is 25 cm. Find the radius of the circle.
4. If tangents PA and PB from an external point P to a circle with center O are inclined to each other at an angle of 80 degrees, then find angle POA.
5. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.