Tangent–Radius Perpendicular Theorem

A tangent is a straight line drawn from an external point that touches a circle at exactly one point on its circumference. Tangents have certain properties that are used in solving mathematical problems related to circles.

Theorem

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

Given: In the figure, PL is a tangent to the circle S with center O.
The tangent touches the circle at point A.

To prove: OA is perpendicular to the tangent PL (OA ⟂ PL).

Proof

Point A is the point of contact of the tangent PL with the circle.
Take any point P on the tangent PL, other than point A, and join OP.

Since point P lies outside the circle, the distance OP > OA, because OA is the radius of the circle.

Thus, among all the distances from O to the points on line PL, OA is the shortest distance.

The shortest distance from a point to a line is the perpendicular.

Therefore, OA ⟂ PL.

Sample Problems based on the Theorem

Problem 1: Given a circle with center O. Two tangents from an external point P are drawn to the given circle. Find the sum of angles formed between both radii and the angles between both the tangents of the circle. 

Solution:

The angles formed between the tangents and the radii is 90 degree. 

Since the sum of angles of the quadrilaterals is 360 degrees. And we have two 90 degrees angles formed within it.
Hence the remaining sum of angles i.e sum of angles formed between both radius and the angles between both the tangents is 360-180=180 degrees. 

Problem 2: Find the angle ∠CBA given that CA is a line drawn from the center to the tangent on the circle. The length of the radius and the base length are mentioned in the question. 

Solution:

It is given that the line is from the center to the tangent, so we conclude that it is at a right angle from the theorem. 

Hence, we can apply trigonometric formulas to get the ∠CBA

tan(∠CBA) = CA/AB 
tan(∠CBA) = 3/4 

∠CBA=37 degree

Problem 3: In the figure given, O is the center of the circle. From point R outside the circle, as shown, RM and RN are tangent, touching the circle at M and N. If the length of OR = 10 cm and the radius of the circle = 5 cm, then what is the length of each tangent? 

Solution:

Given that OM = 5 cm and OR = 10 cm 
In right ∆OMR, 

OR ² = OM ²+ MR ² 
MR ²= OR ² − OM ² 
MR ² = 100 − 25 
MR = 5√3 cm. 
Also NR = 5√3 cm. 

Hence the length of each tangent is 5√3 cm.

Articles Related to the Above Theorem:

• Circles
• Tangent
• Circles Theorem
• Tangent-Secant Theorem