Circles | CBSE Class 10 Revision & Important Questions

Let’s take every topic of Circles one-by-one

When we have a circle and a line, there can be 3 cases:

1. The line will not intersect the circle.
2. It will intersect the circle at 2 different points. In this case, the line is called secant. The line segment inside the circle, i.e., ?? over here is called chord of the circle.
3. The line will touch the circle at one point. Such a line is called tangent of the circle.

Properties of Tangent to Circles

Theorem 1: We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof:

Given: A circle with centre ? and a tangent ?? to the circle at a point ?. To prove: that OP is perpendicular to XY.

Take a point ? on ?? other than ? and join ??.

The point ? must lie outside the circle.

∴ OQ is longer than the radius ?? of the circle, i.e.,

??>??.

Since this happens for every point on the line ?? except the point ?,?? is the shortest of all the distances of the point ? to the points of ??.

So, ??⊥??

Note:
I. This is a very important theorem and its proof has been asked in a lot of previous year boards papers. Do revise it thoroughly before going for the exam.
II. By theorem above, we can also conclude that at any point on a circle there can be one and only one tangent.
III. The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point

Theorem 2: The length of tangents drawn from an external point to a circle are equal.

Proof:
Given: A circle with centre ?, a point ? lying outside the circle and two tangents ??,?? on the circle from ?.
To prove: ??=??
Join OP, OQ and OR.
In right trangles ??? and ???,
∠???=∠???=90° [Radius ⊥ Tangent]
??=??
??=?? [Radii of circle]
∴Δ???≅Δ??? [by ??? congruency criterion]
⇒??=?? [by ????]

Note:
I. This is a very important property and has been asked in 8 out 10 previous years boards papers.
II. As ∠???=∠???, we can say that ?? is the angle bisector of ∠???, i.e., the centre lies on the bisector of the angle between the two tangents.

Let’s solve one of the questions: