Ex.10.2 Q1 Circles Solution - NCERT Maths Class 10

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Question

From a point \(Q,\) the length of the tangent to a circle is \(\text{24 cm} \) and the distance of \(Q\) from the center is \(\text{25 cm.} \)

The radius of the circle is

(A)    \(\text{7 cm}\)         

(B)     \(\text{12 cm}\)          

(C)     \(\text{15 cm}\)           

(D)     \(\text{24.5 cm}\)

Text Solution

What is Known?

(i) Length of tangent from a point \(Q,\) i.e. \(PQ\) is \(\text{24 cm}\)

(ii) Distance of \(Q\) from center, i.e. \(OQ\) is \(\text{25 cm.}\)

What is Unknown?

Radius of the circle

Reasoning:

Tangent at any point of a circle is perpendicular to the radius through the point of contact.

Steps:

\(\therefore \; \Delta \,{OPQ}\) is a right-angled triangle

By Pythagoras theorem,

\[\begin{align} {O Q} ^ { 2 } &= {O P} ^ { 2 } + {P Q} ^ { 2 } \\ 25 ^ { 2 } &= r ^ { 2 } + 24 ^ { 2 } \\ r ^ { 2 } &= 25 ^ { 2 } - 24 ^ { 2 } \\ & = 625 - 576 \\ r ^ { 2 } &= 49 \\ r& = \pm 7 \end{align}\]

Radius cannot be a negative value,

\(\therefore\; r = + 7 \; \rm{cm}\)

Hence the correct  Option is A