UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

Learn more### Theory:

*Result \(3\)*:

The lengths of the two tangents drawn from an exterior point to a circle are equal.

**:**

*Explanation*The lengths of the two tangents \(PA\) and \(PB\) drawn from an exterior point \(P\) to a circle are equal.

\(\Rightarrow\) \(PA = PB\).

Proof for the result:

By the result \(1\), we have:

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

\(OB\) \(\perp\) \(PB\) and \(OA\) \(\perp\) \(PA\).

Here, \(OA\) and \(OB\) are radius. Hence, they are equal.

The side \(OP\) is a common side to the triangles \(AOP\) and \(BOP\).

Thus, the triangles \(AOP\) and \(BOP\) are congruent.

Therefore, \(PA = PB\).

Example:

In the above given figure if \(OB\) \(=\) \(3\) \(cm\) and \(OP\) \(=\) \(5\) \(cm\), find the length of \(PA\).

**Solution**:

By the

**result**\(1\), we have:A tangent at any point on a circle and the radius through the point are perpendicular to each other.

\(\angle OPB\) \(=\) \(90^{\circ}\).

So, \(OPB\) 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.

\(OP^2\) \(=\) \(OB^2\) \(+\) \(PB^2\)

\(PB^2\) \(=\) \(OP^2\) \(-\)\(OB^2\)

\(PB^2\) \(=\) \(5^2\) \(-\) \(3^2\)

\(PB^2\) \(=\) \(25 - 9\)

\(PB^2\) \(=\) \(16\)

\(\Rightarrow\) \(PB\) \(=\) \(\sqrt{16}\)

\(PB\) \(=\) \(4\)

Thus, the measure of \(PB\) \(=\) \(4\) \(cm\)

By the

**result**\(3\), we have:The lengths of the two tangents drawn from an exterior point to a circle are equal.

Hence, \(PA\) \(=\) \(PB\).

Therefore, the measure of \(PA\) \(=\) \(4\) \(cm\)