# Chapter Notes: Circle CBSE Class 10

Notes for circles chapter of class 10 Mathematics. Dronstudy provides free comprehensive chapterwise class 10 Mathematics notes with proper images & diagram.

**(1) Prove that Tangent to a circle at a point is perpendicular to the radius through the point of contact.
**

**Given:**A circle and a tangent at a point .

**To Prove:**

**Construction:**Take any Point , other than , on the tangent . Join . Suppose meets the circle at .

**Proof:**We know that among all line segments joining the point to a point on the shortest one is perpendicular to . So, to prove that , it is sufficient to prove that is shorter than any other segment joining to any point of .

Cleraly,Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Radii of the same circle]

Now, Â Â Â Â Â Â Â Â

Â Â Â Â Â Â OR" />

Â Â Â Â Â Â OP" />Â Â Â Â Â Â Â []

Â Â Â Â Â Â

Thus, is shorter than any other segment joining to any point of .

Hence, .

**(2) Prove that from a point, lying outside a circle, two and only two tangents can be drawn to it.
**When the point lies outsides the circle, there are exactly two tangents to circle from a point which lies outside the circle. As shown in figure.

**(3) Prove that the lengths of the two tangents drawn from an external point to a circle are equal.
**

**Given:**and are two tangents from a point to a circle .

**To Prove:**

**Construction:**Join and

**Proof:**In order to prove that we shall first prove that .

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

and

....(i)

Now, in right triangle and , we have

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Radii of a circle]

Â Â Â Â Â Â [From (i)]

And, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Common]

So, by RHS-Criterion of congruence, we get

Â Â Â Â