Question

In the adjoining 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.

i. Find the length of segment MT. 

ii. Find the length of seg MN. 

iii. Find the measure of ∠NSM.

Answer 1

i. MT = 9 cm [Radius of the bigger circle] 

ii. MT = MN + NT [M – N – T] 

∴ 9 = MN + 2.5 

∴ MN = 9 – 2.5 

∴ MN = 6.5 cm 

iii. seg MR is a tangent to the smaller circle and NS is its radius.

∴ ∠NSM = 90° [Tangent theorem] 

iv. In ∆NSM, ∠NSM = 90° 

∴ MN² = NS² + MS² [Pythagoras theorem] 

∴ 6.5² = 2.5² + MS² 

∴ MS² = 6.5² – 2.5² = (6.5 + 2.5) (6.5 – 2.5) [∵ a² – b² = (a + b) (a – b)] 

= 9 × 4=36 

∴ MS = √36 [Taking square root of both sides] 

= 6 cm 

But, MR = MS + SR [M – S – R] 

∴ 9 = 6 + SR 

∴ SR = 9 – 6

∴ SR = 3cm 

Now, = MS/SR = 6/3 = 2/1 

∴ MS/SR = 2 : 1