Question

proof tangent segment theorem

Answer 1

Given a point outside a circle, two lines can be drawn through that point that are tangent to the circle. The tangent segments whose endpoints are the points of tangency and the fixed point outside the circle are equal. In other words, tangent segments drawn to the same circle from the same point (there are two for every circle) are equal. ,

Answer 2

To find :

Proof of tangent segment theorem.

Solution :

Tangent segment theorem :

A line on the circle is a tangent to that circle if and only if that line is perpendicular to the radius of the circle drawn on the point of incidence.

The properties are as follows:

• The tangent line does not crosses the circle, it just touches the circle.
• At the tangency point, it is perpendicular to the radius.
• A chord and tangent form an angle and this angle is the same as that of tangent inscribed on the opposite side of the chord.
• From that external point, the tangent segments to a circle are equal.

Proof :

We made two tangents from the same external point,

the lengths of both tangents are same.

Lets make a line from that external point to center of circle.

We know that sum of angles between center of circle and external point of a quadrilateral is equal to 180°

As the sum of internal angle in quadrilateral = 360°

So, the sum of angles made by tangents :

$360 \degree \: - \: 180 \degree = 180 \degree$

As

AC = BC

OC = OC

and

angle COA = angle COB

so, by SAS congruancy rule,

Triangle AOC = Triangle BOC

so,

Angle CAO = Angle CBO

As the sum of angles made by tangents = 180°

so

Angle CAO + Angle CBO = 180 °

and

because of CPCT

$2 \times \angle CAO \: = 180 \degree \\ \\ so \\ \angle CAO \: = 90 \degree$

Similarly

$\angle CBO \: = 90 \degree$

This proves that

Tangents made by external point on circle is perpendicular to radius.