Question

In the adjoining figure, circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Prove that, radius XA || radius YB. Fill in the blanks and complete the proof.

Answer 1

Given: X and Y are the centres of circle. 

To prove: radius XA || radius YB 

Construction: Draw segments XZ and YZ.

Proof: 

By theorem of touching circles, points X, Z, 

Y are collinear. 

∴ ∠XZA ≅ ∠BZY [Vertically opposite angles] 

Let ∠XZA = ∠BZY = a …………….. (i) 

Now, seg XA seg XZ [Radii of the same circIe] 

∴ ∠XAZ ≅∠XZA = a …………….. (ii) [Isosceles triangle theorem]

Similarly, seg YB ≅ seg YZ [Radii of the same circie] 

∴ ∠BZY = ∠ZBY = a …………….. (iii) [Isosceles triangle theorem] 

∴ ∠XAZ = ∠ZBY [From (i), (ii) and (iii)] 

∴ radius XA || radius YB [Alternate angles test]