Q6 of 59 Page 83

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.



(1)MT = radius of the big circle = 9 cm


(2)MN = MT – TN = 9 – 2.5 = 6.5 cm


(3)SM is the tangent to the circle with radius 2.5 cm with S being point of contact.


∠ NSM = 90° Using tangent-radius theorem which states that a tangent at any point of a circle is perpendicular to the radius at the point of contact.


In ∆MSN,


∠ MSN = 90°{∵ MS is the tangent to the small circle with point of contact S}


⇒ MN2 = MS2 + NS2


MS2 = MN2 – NS2


⇒ MS2 = 6.52 – 2.52


⇒ MS2 = 36


⇒ MS = 6 cm


Now, SR = MR – MS = 9 – 6 = 3 cm


⇒ MS:SR = 6:3 = 2:1


More from this chapter

All 59 →
4

In figure 3.84, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle isr and l(AB) = r , Prove that, is a square.


 5

In figure 3.85, is a parallelogram. It circumscribes the circle with cnetre T. Point E,F, G, H are touching points. If AE = 4.5,EB = 5.5, find AD.


 7

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........)


 8

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.