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 CBSE Maths eBooks CBSE Guess > eBooks > Class X > Maths Part II by Mr. M. P. Keshari 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]   Maths by Mr. M. P. Keshari