Question

In the given figure, 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

Radius of circle with centre M = 9 cm

Radius of circle with centre N = 2.5 cm

Join MT and NS.



If two circles touch each other, their point of contact lie on the line joining their centres. So, the points M, N and T are collinear.

(1)
Length of segment MT = 9 cm (Radius of circle with centre M)

Thus, the length of the segment MT is 9 cm.

(2)
Length of segment NT = 2.5 cm (Radius of circle with centre N)

∴ Length of segment MN = Length of segment MT − Length of segment NT = 9 − 2.5 = 6.5 cm

Thus, the length of the segment MN is 6.5 cm.

(3)
The tangent at any point of a circle is perpendicular to the radius through the point of contact.

In the given figure, seg RM is tangent to the circle with centre N at point S.

∴ ∠NSM = 90º

In right ∆NSM,

$$\begin{gathered} {\mathrm{MN}}^2={\mathrm{NS}}^2+{\mathrm{SM}}^2 \\ \Rightarrow \mathrm{SM}=\sqrt{{\mathrm{MN}}^2-{\mathrm{NS}}^2} \\ \Rightarrow \mathrm{SM}=\sqrt{{\left(6.5\right)}^2-{\left(2.5\right)}^2} \\ \Rightarrow \mathrm{SM}=\sqrt{42.25-6.25} \\ \Rightarrow \mathrm{SM}=\sqrt{36}=6\mathrm{cm} \end{gathered}$$
∴ SR = MR − SM = 9 − 6 = 3 cm (MR = Radius of the circle with centre M)

⇒ MS : SR = 6 cm : 3 cm = 2 : 1

Thus, the ratio MS : SR is 2 : 1.