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*Result \(1\)*:

A tangent at any point on a circle and the radius through the point are perpendicular to each other.

**:**

*Explanation*The tangent at the point \(P\) on a circle and the radius through the point \(P\) are perpendicular.

That is, the radius \(OP\) makes an angle \(90^{\circ}\) with the tangent \(AB\) at the point \(P\).

Example:

In the above given figure if \(OP\) \(=\) \(3\) \(cm\) and \(PQ\) \(=\) \(4\) \(cm\), find the length of \(OQ\).

**Solution**:

By the result, \(\angle OPQ\) \(=\) \(90^{\circ}\).

So, \(OPQ\) is a right-angled triangle.

By the Pythagoras theorem, we have:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

\(OQ^2\) \(=\) \(OP^2\) \(+\) \(PQ^2\)

\(OQ^2\) \(=\) \(3^2\) \(+\) \(4^2\)

\(OQ^2\) \(=\) \(9 + 16\)

\(OQ^2\) \(=\) \(25\)

\(\Rightarrow\) \(OQ\) \(=\) \(\sqrt{25}\)

\(OQ\) \(=\) \(5\)

Therefore, the measure of \(OQ\) \(=\) \(5\) \(cm\)

*Result \(2\)*:

- No tangent can be drawn from an
**interior point of the circle**.

- Only one tangent can be drawn at any
**point on a circle**.

- Two tangents can be drawn from any
**exterior point of a circle**.