CBSE Maths eBooks
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)
Construction:- A diameter AOE is drawn. BE is joined.
Proof: - In
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
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)
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.