In figure 3.88, circles with centres X and Y touch internally at point Z. Seg BZ is a chord of bigger circle and intersects smaller circle at point A. Prove that, seg AX || seg BY.

In figure 3.86, circle with centre M touches the circle with centre N at point T. Radius RM touches the smaller circle at S. Radii of circles are 9 cm and 2.5 cm. Find the answers to the following questions hence find the ratio MS:SR.

(1) Find the length of segment MT

(2) Find the length of seg MN

(3) Find the measure of ∠ NSM.

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.

Construction: Draw segments XZ and ..YZ........ .

Proof: By theorem of touching circles, points X, Z, Y are ..concyclic........ .

..∠ YZB ........vertically opposite angles

Let ..... (I)

Now, seg XA ≅seg XZ ........ (...radius of the same circle.......)

= ....∠ XZA...... = a ........ (isosceles triangle theorem) (II)

similarly, seg YB ≅ .YZ......... ........ (.radius of the same circle.........)

.∠ZBY......... = a ........ (.isosceles triangle theorem.........) (III)

∴from (I), (II), (III),

∠ XAZ = .∠ ZBY.........

∴radius XA || radius YB .......... (..since alternate interior angles are equal........)

In figure 3.89, line l touches the circle with centre O at point P. Q is the mid point of radius OP. RS is a chord through Q such that chords RS || line l. If RS = 12 find the radius of the circle.

In figure 3.90, seg AB is a diameter of a circle with centre C. Line PQ is a tangent, which touches the circle at point T. seg AP line PQ and seg BQ line PQ. Prove that, seg CP ≅seg CQ.