Chapter 10 : Tangents to a circle

Theorem 3. If a line touches a circle and from the point of contact a chord is drawn, the angle which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments.

Given:- PQ is a tangent to circle with centre O at a point A, AB is a chord and C, D are points in the two segments of the circle formed by the chord AB.

To Prove:- (i)

                      (ii)

Construction:- A diameter AOE is drawn. BE is joined.

Proof: - In

From Theorem 3,

Again [Linear Pair]

and       [Opposite angles of a cyclicquad]

       

Theorem 4. If a line is drawn through an end point of a chord of a circle so that the angle formed by it with the chord is equal to the angle subtend by chord in the alternate segment, then the line is a tangent to the circle.

Given:- A chord AB of a circle and a line PAQ. Such that where c is any point in the alternate segment ACB.

To Prove:- PAQ is a tangent to the circle.

Construction:- Let PAQ is not a tangent then let us draw P' AQ' another tangent at A.

Proof: - AS P ’AQ’ is tangent at A and AB is any chord

[theo.3]

But (given)

Hence AQ' and AQ are the same line i.e. P' AQ' and PAQ are the same line.

Hence PAQ is a tangent to the circle at A.

Theorem 5. If two circles touch each other internally or externally, the point of contact lie on the line joining their centres.

Given:- Two circles with centres O₁ and O₂ touch internally in figure (i) and externally in figure (ii) at A.

To prove: - The points O₁, O₂ and A lie on the same line.

Construction:- A common tangent PQ is drawn at A.

Proof: - In figure (i) (PA is tangent to the two circles)

O₁, O₂ and A are collinear.

In figure (ii) (PA is tangent to the circles)

i.e. from a linear pair

O₁, O₂ and A lie on the same line.

Example 5. In the given figure TAS is a tangent to the circle, with centre O, at the point A. If , find the value of x and y.

Solution:-

Now TAS is tangent at A

                   [Angles in the alternateseg]