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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. |
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Answer» 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] |
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