Q. 64.3( 24 Votes )

# 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.

Answer :

(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}

⇒ MN^{2} = MS^{2} + NS^{2}

MS^{2} = MN^{2} – NS^{2}

⇒ MS^{2} = 6.5^{2} – 2.5^{2}

⇒ MS^{2} = 36

⇒ MS = 6 cm

Now, SR = MR – MS = 9 – 6 = 3 cm

⇒ MS:SR = 6:3 = 2:1

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