medium difficulty5 min

Radius perpendicular to tangent

The radius drawn to the point of contact of a tangent is perpendicular to the tangent.

In one line: Radius and tangent make a right angle at the touch point.

Understand it

This is one of the most important tangent properties in Class 10. If a tangent touches the circle at point P, then the radius from the centre to P makes a 90 degree angle with the tangent. This result is often used in proofs and angle questions.

Example

If OA is a radius and line l is tangent at A, then OA is perpendicular to l.

Formula and rule

Radius to the point of contact is perpendicular to the tangent: OA ⟂ tangent at A.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1If a circle has centre O and a tangent touches it at A, then OA is drawn.
2The angle between OA and the tangent is 90 degrees.

Method you can reuse

1Identify the point of contact.
2Draw the radius from the centre to that point.
3Use the property that radius and tangent make a right angle.
4Write the angle as 90 degrees in the solution.

Sketch it in your answer

Draw the centre, the point of contact, and a small right-angle square at the contact point.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A tangent touches a circle at A. What is the angle between the tangent and the radius drawn to A?

Think first, then show answer

The angle is 90 degrees. The radius to the point of contact is always perpendicular to the tangent.

Common wrong answer

A common wrong answer is 0 degrees or 180 degrees.

Why it loses marks: The radius does not lie along the tangent line; it meets it at right angles.

Don't mix this up

Students often write that the radius is parallel to the tangent, which is opposite to the actual theorem.

Exam tip

Whenever you see a tangent and radius meeting at the contact point, mark a right angle immediately.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

This property is asked in angle-finding questions and short proofs from a circle diagram.

1
Enough for 1 mark
The radius drawn to the point of contact of a tangent is perpendicular to the tangent.
2
Add for 2 marks
Example: If OA is a radius and line l is tangent at A, then OA is perpendicular to l.
3
Add for 3 marks
If a tangent touches the circle at point P, then the radius from the centre to P makes a 90 degree angle with the tangent.

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