CBSE Class 10 Mathematics Circles Notes Set B

CIRCLES

BASIC CONCEPTS RELATED TO CHAPTER

You might have seen many objects that are circular in shape.
· A circle is defined as the collection of all the points on a plane that are at equal distance from a given fixed point on the plane.
· This fixed point is called centre of the circle and the fixed distance is called the radius.
· A line segment joining the centre of a circle with any point on its circumference is called the radius of the circle.
· A line that joins two points on the circumference of a circle is called a chord.
· A chord that passes through the centre of a circle is called the diameter of the circle.
· A diameter divides a circle into two equal parts, each is called a semicircle. Diameter is
the longest chord of a circle. The diameter of a circle is twice the radius.
· The tangent to a circle is a line that touches the circle at one point.
L is the point where the tangent touches the circle.It is called the point of contact.

· Secant intersects the circle at two points.

· The point at which the straight line touches the circle is called the point of contact or point of tangency.

Some properties of tangents to a circle are:-
1. Infinite number of tangents can be drawn to a circle but only one tangent can be drawn at any given point on a circle

2. From an external point we can draw two tangents of equal length

3. The radius of the circle is perpendicular to the tangent at its point of contact and the tangents drawn at the extremities of the diameter of a circle are parallel.

THEOREMS

Theorem 1:-The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact.

Given: A tangent AB with point of contact P.
To prove: OP ⊥ AB
Proof: Consider point C on AB other than P.
C must lie outside the circle. (∵A tangent can have only one point of contact with the circle)
OC > OP (∵ C lies outside the circle)
This is true for all positions of C on AB.
Thus, OP is the shortest distance between point P and line segment AB.
Hence, OP ⊥ AB.

Theorem 2:Tangents drawn to a circle from an external point are equal in length.

Given:- Two tangents AB and AC from an external point A to points B and C on a circle.
To prove: AB = AC
Construction: Join OA, OB and OC.
Proof:In triangles OAB and OAC,
∠OBA = 90⁰ (Radius OB ⊥ Tangent AB at B)
∠OCA = 90⁰ (Radius OC ⊥ Tangent AC at C)
In triangles OBA and OCA,
∠OBA = ∠OCA = 90⁰
OB = OC (Radii of the same circle)
OA = OA (Common side)
Thus, ΔOBA =̃ΔOCA (RHS congruence rule)
Hence, AB = AC (By C.P.C.T)