**Theorem 10.1**

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Tangent -> It is nothing but a line which passes through a curve only at one point.

Note: circle is also a curve.

By only at one point, we mean that this line touches the circle at one point and if we extend this line, this line will never touches any other point.

Note: The line which touches the circle at two points is called secant.

We can also find tangent when the two points of the secant come nearer and nearer until they become one.

To know how this theorem came, let’s visualise a situation.

Suppose ram is standing on the footpath and he wants to cross the road and go to the other side.

Now, if he walks to the point P, it will take more time due to longer distance.If he walks to the point Q, It will take less time.Now, if he wants to go to the other side as soon as possible, he has to go straight from A to B. It just means that the shortest distance between two lines is always perpendicular.

Now, if we take some points on the tangent to the circle and join these points to the center of the circle, we observe that OP is longer than OQ. And the shortest is OM. So, OM is perpendicular to the tangent.

Now, M is nothing but that point where the tangent touches the circle.

This is the proof of the above theorem.

**Theorem 10.2**

The lengths of tangents drawn

from an external point to a circle are equal.

Do you know that we have only two tangents from an external point on a given circle.

The proof of the above theorem is simple.

We know that two triangles are said to be congruent if all the three sides and three angles of one triangle are equal to the other’s.

And to prove the congruency of them, we have SSS, SAS, ASA, AAS rules.

We also have a rule for particularly right angled triangles called HL rule:

Two right triangles are congruent if the hypotenuse and one corresponding leg are equal in both triangles.

Now if we see the following figure,

we have

OQ= OR ( Radius)

OP = OP ( Common)

Hence, these two triangles are congruent.

So, PQ = PR.

Hence proved.

Good day! boundtoteach.com

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