Tangents to a Circle and The Radius-Tangent Perpendicularity for CBSE Class 10

In Class 9, you explored the fundamental properties of circles—chords, arcs, subtended angles, and cyclic quadrilaterals. But what happens when a straight line and a circle lie in the exact same two-dimensional plane? Depending on their relative positions, the line might miss the circle entirely, slice through it at two distinct points, or touch the curved boundary at strictly one single point.

In CBSE Class 10 Mathematics, Chapter 10 (Circles) shifts focus to this last, exquisite geometric phenomenon: the tangent to a circle. At the heart of circle geometry lies Theorem 10.1—the foundational principle establishing that the radius at the point of contact is always strictly perpendicular to the tangent line.

What You Will Learn

• A straight line and a circle: Non-intersecting lines, Secants, and Tangents
• Formal definition of a tangent and the point of contact
• How many tangents can be drawn to a circle at a given point?
• Parallel tangents to a circle: Why at most two parallel tangents exist along any diameter
• Rigorous step-by-step geometric proof of Theorem 10.1 (Radius-Tangent Perpendicularity)
• Calculating unknown lengths using the Pythagoras Theorem in right-angled tangent triangles
• Solved CBSE board examination problems, presentation rules, and common mistakes

1. A Line and a Circle in a Plane

Consider a circle with center $O$ and a straight line $PQ$ lying in the same plane. There are only three mutually exclusive geometric possibilities:

    Case 1: Non-Intersecting Line     Case 2: Secant                  Case 3: Tangent
               O                               O                               O
                                                                               |
          +---------+                     +----+----+                     +----+----+
          |  Circle |                     |    |    |                     |    P    |
          +---------+                     +----+----+                     +----+----+
                                               |                               |
    P---------------------Q              P-----+-----Q                  P------+------Q
     (No common points)                  (2 common points: A, B)        (Exactly 1 common point: P)

1. Non-Intersecting Line: The line $PQ$ and the circle have no common points.
2. Secant: The line $PQ$ intersects the circle at two distinct points ($A$ and $B$).
3. Tangent: The line $PQ$ touches the circle at strictly one single point ($P$).

2. What is a Tangent to a Circle?

Formal Definition

A tangent to a circle is a straight line that touches or intersects the circle at only one point. The unique single point where the tangent touches the circle is called the point of contact.

Key Geometric Properties of Tangents:

1. Uniqueness at a Point: At any given point on the circumference of a circle, there is one and only one tangent that can be drawn.
2. Infinite Tangents to a Circle: Since a circle contains an infinite number of points on its circumference, a circle possesses an infinite number of tangents.
3. Parallel Tangents along a Diameter: <u>A circle can have at most TWO parallel tangents at any given time, and these two parallel tangents must pass through the opposite endpoints of a diameter!</u>

3. Theorem 10.1: The Radius-Tangent Perpendicularity Theorem

Theorem Statement (CBSE Theorem 10.1)

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

                                      O (Center)
                                     /|
                                    / |
                                   /  | Radius r
                            OQ > r/   |
                                 /    | (90°)
    X ------------------------- Q ----P------------------------ Y (Tangent)
                                  Point of Contact

Given:

A circle with center $O$ and a tangent line $XY$ touching the circle at point $P$.

To Prove:

$OP \perp XY$

Construction:

Take any point $Q$ on the tangent line $XY$, other than the point of contact $P$. Join $OQ$.

Step-by-Step Proof by Contradiction:

1. Location of Point $Q$: The point $Q$ must lie outside the circle.
   • Reason: If point $Q$ were to lie inside the circle, then line $XY$ would intersect the circle at two points and become a secant, not a tangent!
   • Because $XY$ is a tangent, every point on $XY$ (except point $P$) lies outside the circle.
2. Comparing Distances: Since point $Q$ lies outside the circle, the distance $OQ$ from the center must be strictly greater than the radius of the circle $OP$: $OQ > OP$
3. Generalization for All Points on Line $XY$: This inequality ($OQ > OP$) holds true for every single point on the line $XY$, except the point of contact $P$.
4. The Geometric Principle: Therefore, $OP$ is the shortest distance from the center $O$ to any point on the line $XY$.
5. Conclusion: From Euclidean geometry, the shortest distance between a given point and a straight line is always the perpendicular distance. Therefore: $\mathbf{OP \perp XY}$ Hence, proved.

Important: <u>In any problem involving a circle and a tangent, the moment you draw a radius to the point of contact, you immediately create a $90^\circ$ right angle (ngle OPT = 90^\circ). This allows you to apply the Pythagoras Theorem to solve for unknown lengths!</u>

4. Solved CBSE Board Examination Problems

Solved Example 1: Calculating Tangent Length Using Pythagoras

Problem: A tangent $PQ$ at a point $P$ of a circle of radius $5\text{ cm}$ meets a line through the center $O$ at a point $Q$ so that $OQ = 12\text{ cm}$. Find the length of the tangent $PQ$.

Solution:

1. Analyze the Geometry:
   • Radius: $OP = 5\text{ cm}$.
   • Distance from center: $OQ = 12\text{ cm}$.
   • By Theorem 10.1, the radius through the point of contact is perpendicular to the tangent: $OP \perp PQ \implies \angle OPQ = 90^\circ$
2. Apply Pythagoras Theorem in $\Delta OPQ$: Here, $OQ$ is the hypotenuse: $OQ^2 = OP^2 + PQ^2$ $12^2 = 5^2 + PQ^2$ $144 = 25 + PQ^2$
3. Solve for $PQ$: $PQ^2 = 144 - 25 = 119$ $PQ = \mathbf{\sqrt{119}\text{ cm}}$
4. Therefore, <u>the length of the tangent $PQ$ is $\sqrt{119}\text{ cm}$</u>.

Solved Example 2: Proving Tangents at Diameter Endpoints Are Parallel

Problem: Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Solution:

1. Let $AB$ be a diameter of a circle with center $O$.
2. Let line $PQ$ be the tangent at endpoint $A$, and line $RS$ be the tangent at endpoint $B$.
3. By Theorem 10.1, radius $OA \perp PQ$: $\angle PAB = 90^\circ \quad \text{--- (1)}$
4. Similarly, radius $OB \perp RS$: $\angle ABS = 90^\circ \quad \text{--- (2)}$
5. Notice that for straight lines $PQ$ and $RS$ with transversal line $AB$: $\angle PAB = \angle ABS = 90^\circ$
6. Since these two angles are alternate interior angles and are equal, the lines must be parallel: $\mathbf{PQ \parallel RS}$
7. Hence, proved.

5. Summary and Examination Tips

Exam Tip: In right-angled tangent triangles, remember that the distance from the center to the external point ($OQ$) is ALWAYS the hypotenuse because it lies opposite the $90^\circ$ angle at the point of contact! Do not make the common mistake of setting the tangent as the hypotenuse.

Common Mistake: Omitting the reason "radius is perpendicular to the tangent at the point of contact" when using a right angle in proof questions. Stating the theorem in brackets is worth half a mark in the marking scheme!