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 (1) MT (2) MN (3) ∠NSM, hence find the ratio MS:SR.

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 \(\angle NSM\).

Figure: The smaller circle (centre N) lies inside the bigger circle (centre M) and touches it internally at T. R is on the bigger circle and S lies on radius MR.

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Answer: (1) 9 cm (2) 6.5 cm (3) \(90^\circ\); MS : SR = 2 : 1

(1) T is on the bigger circle, so MT is its radius: MT = 9 cm.

(2) The circles touch internally, so

\[MN = 9 - 2.5 = 6.5\ \text{cm}\]

(3) MR touches the smaller circle at S, so radius NS ⊥ MR: \(\angle NSM = 90^\circ\).

In right-angled \(\triangle NSM\):

\[MS^2 = MN^2 - NS^2 = 42.25 - 6.25 = 36\] \[MS = 6\ \text{cm},\quad SR = 9 - 6 = 3\ \text{cm}\] \[MS : SR = 6 : 3 = 2 : 1\]

Doubtly AI · 24 Oct 2026 AI-verified · Oct 2026 Report

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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 (1) MT (2) MN (3) ∠NSM, hence find the ratio MS:SR.

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