Question

usic
B
Y
Z
In the figure, circles with centres X and Y touch internally at point Z. seg BZ
is a chord of bigger circle and it intersects smaller circle at point A. Prove that
seg AX seg BY.​

Answer 1

Answer:

Circles with centres X and Y touch internally point Z.

Join YZ.

(Refer image)

By theorem of touching circles, points Y, X, Z are collinear.

Now, segXA≅segXZ (Radii of circle with centre X)

∴∠XAZ=∠XZA (Isosceles triangle theorem) (1)

Similarly, segYB≅segYZ (Radii of circle with centre Y) ∴∠BZY=∠ZBY (Isosceles triangle theorem) (2) From (1) and (2), we have ∠XAZ=∠ZBY

If a pair of corresponding angles formed by a transversal on two lines is congruent, then the two lines are parallel.

∴segAX∣∣segBY (Corresponding angle test)