# 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

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