medium difficulty5 min

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.

In one line: Touch point plus radius equals right angle.

Understand it

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.

Example

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

Formula and rule

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

See it worked out

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

Worked example

1In triangle OAT, if AT is tangent at A, then angle OAT = 90 degrees.
2This can then help find the remaining angles.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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

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.

Don't mix this up

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

Exam tip

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

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

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

1
Enough for 1 mark
At the point where a tangent touches a circle, the radius to that point makes a right angle with the tangent.
2
Add for 2 marks
Example: If OA is a radius and AT is tangent at A, then angle OAT is 90 degrees.
3
Add for 3 marks
This property helps in angle chasing.

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