A tangent to a circle is defined as a line segment that touches the circle exactly at one point. 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.
- There is exactly one tangent to a circle which passes through only one point on the circle.
- There are exactly two tangents can be drawn to a circle from a point outside the circle.
In the figure, \(P\)
Some theorems on length of tangen
Theorem 1: The lengths of tangents drawn from an external point to a circle are equal. It is proved as follows:
Consider the circle with center \(O\)
Since tangent on a circle and the radius are perpendicular to each other at the point of tangency,
Consider the triangles, \(∆PAO\)
Hence, by RHS congruence theorem,
\(∆PAO ≅ ∆PBO\)
\(⇒ PA = PB\)
This can also be proved by using Pythagoras theorem as follows,
This gives, \(PA\)
Therefore, tangents drawn to a circle from an external point will have equal lengths. There is an important observation here:
- Since \(∠APO\)
= \(∠BPO\) , \(OP\) is the angle bisector of \(∠APB\) .
Therefore, center of the circle lies on angle bisector of the angle made by two tangents to the circle from an external point.
Let’s consider an example for better understanding of the concept of length of tangent drawn to a circle from an external point.
Example: A circle is inscribed in the quadrilateral \(ABCD\)
Tangents drawn from the point \(A\)
Similarly, for tangents drawn from point \(B\)
From point \(C\)
From point \(D\)
Adding equations(1),(2), (3) and (4) gives,
\(AP + BN + CN + DP\)
\(\Rightarrow AP + PD + BN + NC\)
\(\Rightarrow AD + BC\)
That’s the required proof.
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