Ex.10.2 Q7 Circles Solution - NCERT Maths Class 10

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Question

Two concentric circles are of radii \(\text{5 cm}\) and \(\text{3 cm.}\) Find the length of the chord of the larger circle which touches the smaller circle.

Text Solution

What is Known?

Two concentric circles are of radii \(\text{5 cm}\) and \(\text{3 cm.}\)

What is Unknown?

The length of the chord of the larger circle which touches the smaller circle.

Reasoning:

Chord of the larger circle is a tangent to the smaller circle.

Steps:

\(PQ\) is chord of a larger circle and tangent of a smaller circle. Tangent is perpendicular to the radius at the point of contact \(S.\)

\(\therefore \;\angle {OSP} = {90^ \circ }\)

In \(\Delta {OSP}\) (Right angled triangle)

By the Pythagoras Theorem,

\[\begin{align} {O P} ^ { 2 } & = {O S} ^ { 2 } + {S P} ^ { 2 } \\ 5 ^ { 2 } & = 3 ^ { 2 } + {S P} ^ { 2 } \\ {S P} ^ { 2 } & = 25 - 9 \\ {S P} ^ { 2 } & = 16 \\ {S P} & = \pm 4 \end{align}\]

\(SP\) is length of tangent and cannot be negative

\(\therefore \;SP = \rm{4\, cm}\)

\(QS = SP\) (Perpendicular from center bisects the chord considering the larger circles)

Therefore, \(QS = SP = \text{4 cm}\)

Length of the chord

\[\begin{align} {PQ} & = \text{QS + SP} \\ & = 4 + 4 \\ { PQ} & = 8 \; \rm{ cm }\end{align}\]

Therefore, the length of the chord of the larger circle is \(\text{8 cm.}\)

  
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