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Radius perpendicular to tangent

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 0 degrees or 180 degrees. That is wrong because the radius does not lie along the tangent line; it meets it at right angles.

Definition

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

Example

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

Rule to remember

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

Memory hook

Radius and tangent make a right angle at the touch point.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

A common wrong answer is 0 degrees or 180 degrees. That is wrong because the radius does not lie along the tangent line; it meets it at right angles.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

The radius drawn to the point of contact of a tangent is perpendicular to the tangent. Radius to the point of contact is perpendicular to the tangent: OA ⟂ tangent at A. If OA is a radius and line l is tangent at A, then OA is perpendicular to l.

3-mark answer

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. Radius to the point of contact is perpendicular to the tangent: OA ⟂ tangent at A. If a circle has centre O and a tangent touches it at A, then OA is drawn. The angle between OA and the tangent is 90 degrees. This property is asked in angle-finding questions and short proofs from a circle diagram. A common wrong answer is 0 degrees or 180 degrees. That is wrong because the radius does not lie along the tangent line; it meets it at right angles.
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