Ex.10.1 Q3. Circles Solution - NCERT Maths Class 10

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Question

A tangent \(PQ\) at a point \(P\) of a circle of radius \(\text{5 cm}\) meets a line through the center \(O\) at a point \(Q\) so that \(OQ = \text{12 cm.}\) Length \(PQ\) is:

(A) \(\text{12 cm}\)        (B) \(\text{13 cm}\)        (C) \(\text{8.5 cm}\)       (D) \(\sqrt{119} \,\rm{cm.}\)

Text Solution

 

What is Known?

Radius \(OP = \text{5 cm} \)

\(OQ = \text{12 cm}\)

What is Unknown?

Length of the tangent \(PQ\)

Reasoning:

\(\Delta {OPQ}\) is a right-angle triangle according to Theorem \(10.1 :\) The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Steps:

By Pythagoras theorem

\[\begin{align} {OQ} ^ { 2 } & = {O P} ^ { 2 } + {P Q} ^ { 2 } \\ 12 ^ { 2 } & = 5 ^ { 2 } + {P Q} ^ { 2 } \\ 144 & = 25 + {P Q} ^ { 2 } \\ {P Q} ^ { 2 } & = 119 \\ {P Q} & = \sqrt { 119 } \rm {\;cm } \end{align}\]

The answer is option D.

  
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