Length Of Tangent On A Circle

A tangent to a circle is defined as a line that touches the circle exactly at one point. The point at which the tangent touches the circle is called the point of contact. Based on the position of this point, we can define the number tangents than can be drawn to a circle. There are some important points regarding tangents:

• A tangent to a circle cannot be drawn through a point which lies inside the circle. It is so because all the lines passing through any point inside the circle will intersect the circle at two points. This can be observed in the below figure.
  
• There is exactly one tangent to a circle which passes through only one point on the circle as shown in the below figure.
  
• There are exactly two tangents can be drawn to a circle from a point outside the circle as shown in the following figure.

In the figure,

Some theorems on length of tangent

Theorem 1: The lengths of tangents drawn from an external point to a circle are equal.

Proof:

Consider the circle with center

Tangent is perpendicular to the radius through the point of contact.

Consider the triangles,

Hence, by RHS congruence theorem,

This can also be proved by using Pythagoras theorem as follows,

Since,

Since

This gives,

Therefore, tangents drawn to a circle from an external point have equal lengths.

There is an important observation here:

• Since
  $\begin{array}{l}\angle APO\end{array}$
  =
  $\begin{array}{l}\angle BPO\end{array}$
  ,
  $\begin{array}{l}OP\end{array}$
  is the angle bisector of
  $\begin{array}{l}\angle APB\end{array}$
  .

Therefore, the centre of the circle lies on the angle bisector of the angle made by two tangents to the circle from an external point.

Also, read:

• Construction of tangent to a Circle
• Tangent – Equation of Tangent and Normal
• Number of Tangent from a Point on a Circle

Let’s consider an example for better understanding of the concept of length of the tangents drawn to a circle from an external point.

Solved Example

Example: A circle is inscribed in the quadrilateral

Solution:

We know that the lengths of tangents drawn from an external point to a circle are equal.

Tangents drawn from the point

Similarly, for tangents drawn from point

Tangents drawn from point

Tangents drawn from point

Adding equations (1),(2), (3) and (4),

By rearranging the terms,

From the given figure,

AM + MB = AB and CO + OD = CD

Hence proved.

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