Question

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

In the given question, we have to find out 4 things : -

1. MS:SR
2. Length of seg MT
3. Length of seg MN
4. Measure of ∠NSM.

Given,

Radius of MT = 9cm

Radius of TN = 2.5cm

Length of MT(Radius) = Radius of RM = 9cm

Length of MN

→ MT - NT

→ 9 - 2.5 = 6.5 cm

Here RS is a tangent of smaller circle. You should remember the theorem i.e tangent at any point of a circle is perpendicular to the radius through the point of contact.

Measure of ∠NSM = 90°

Let's find out the ratio of MS : SR

We need to find out the length of MS and SR

Now, join M to N

In ∆MNS

• Use Pythagoras theorem

→ MN² = NS² + SM²

→ SM² = MN² - NS²

→ SM = √(6.5)² - (2.5)²

→ SM = √42.25 - 6.25

→ SM = √36

→ SM = 6cm

So, the length of SR = MR - SM = 9 - 6 = 3cm

MS:SR = 6 : 3 = 2: 1

MS:SR = 2 : 1

• Length of seg MT = 9cm

Length of seg MN = 6.5 cm

• Measure of ∠NSM. = 90°

Answer 2

QUESTION :-

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.

GIVEN :-

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

TO FIND :-

find the ratio MS:SR = ?

Find the length of segment MT = ?

Find the length of seg MN = ?

Find the measure of ∠NSM = ?

SOLUTION :-

• MT = 9 cm (Radius of bigger circle. When two circles touch each other, the point of contact lies on the line joining centres, M-N-T.)

MT = MN + NT

MN = 9-2.5 = 6.5 cm

Seg MR is tangent to the smaller circle

and seg NS is a radius.

∠NSM = 90°

In ∠NSM , ∠NSM = 90°

M N^ 2 = MS ^ 2 + NS ^ 2

6.5 ^ 2 = MS ^ 2 + 2.5 ^ 2

M S ^ 2 = 42.25 - 6.25

MS ^ 2 = 36

MS = 6 cm

MR = MS + SR

9 = 6 + SR

SR = 9 - 6

SR = 3 cm

MS/SR = 6/3

MS:SR = 2:1