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Using radius-tangent angle relationship

At the point where a tangent touches a circle, the radius to that point makes a right angle with the tangent.

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Student-friendly explanation

This property helps in angle chasing. If one angle in a circle figure depends on the tangent, you can immediately replace it with 90 degrees. Then the rest of the angles often follow by simple angle rules.

How to write this in exams

  1. 1

    Start with the exact idea

    At the point where a tangent touches a circle, the radius to that point makes a right angle with the tangent.

  2. 2

    Then show how to use it

    1. Find the point of contact. 2. Draw or identify the radius to that point. 3. Replace the angle with 90 degrees. 4. Use other angle rules if needed.

  3. 3

    Add one concrete example

    If OA is a radius and AT is tangent at A, then angle OAT is 90 degrees.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to write a random acute angle because the figure looks slanted. The rule is fixed: the angle is 90 degrees.

Definition

At the point where a tangent touches a circle, the radius to that point makes a right angle with the tangent.

Example

If OA is a radius and AT is tangent at A, then angle OAT is 90 degrees.

Rule to remember

Radius at the point of contact is perpendicular to the tangent, so the angle is 90 degrees.

Memory hook

Touch point plus radius equals right angle.

Examples and method

Worked example

In triangle OAT, if AT is tangent at A, then angle OAT = 90 degrees. This can then help find the remaining angles.

Method to apply

1. Find the point of contact. 2. Draw or identify the radius to that point. 3. Replace the angle with 90 degrees. 4. Use other angle rules if needed.

Diagram support

Mark the right angle clearly at the contact point to avoid angle confusion later.

How CBSE asks it

It is often used in angle-sum questions and in proving that a line is tangent.

Avoid common mistakes

Common confusion

Students often treat the angle as unknown and try to calculate it from scratch, even when the tangent rule already gives it.

Common wrong answer

A common wrong answer is to write a random acute angle because the figure looks slanted. The rule is fixed: the angle is 90 degrees.

Exam tip

When a radius meets a tangent at the point of contact, write 90 degrees first and save time.

Quick check

If a radius meets a tangent at the point of contact, what angle do they form?

They form a right angle, so the angle is 90 degrees. This is the standard radius-tangent relationship in circles.

Answer writing and exam use

1-mark answer

At the point where a tangent touches a circle, the radius to that point makes a right angle with the tangent.

2-mark answer

At the point where a tangent touches a circle, the radius to that point makes a right angle with the tangent. Radius at the point of contact is perpendicular to the tangent, so the angle is 90 degrees. If OA is a radius and AT is tangent at A, then angle OAT is 90 degrees.

3-mark answer

This property helps in angle chasing. If one angle in a circle figure depends on the tangent, you can immediately replace it with 90 degrees. Then the rest of the angles often follow by simple angle rules. Radius at the point of contact is perpendicular to the tangent, so the angle is 90 degrees. In triangle OAT, if AT is tangent at A, then angle OAT = 90 degrees. This can then help find the remaining angles. It is often used in angle-sum questions and in proving that a line is tangent. A common wrong answer is to write a random acute angle because the figure looks slanted. The rule is fixed: the angle is 90 degrees.
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