A tangent PQ at a point P of a circle of radius 5 cm meets a line through the center O at a point Q so that OQ = 12 cm. Length of PQ is:
A. \[\sqrt{119}cm\]
B. 13 cm
C. 10 cm
D. 12 cm
VerifiedHint: Draw a rough figure from the conditions given in the question. Use Pythagoras theorem and find the length of PQ.
Complete step-by-step answer:
Consider the figure drawn. PQ represents the tangent drawn to the circle. O is the center of the circle. Thus OP = 5 cm, which is the radius of the circle, and OQ = 12 cm.
From the figure, we can say that OP is perpendicular to the tangent, i.e. \[OP\bot PQ.\]
The tangent at any point of circle is perpendicular to the radius through the point of contact.
\[\therefore \angle OPQ={{90}^{\circ }}\].
Hence, \[\Delta OPQ\]forms a right angled triangle.
We need to find the length of PQ.
Consider \[\Delta OPQ\], right angled at P.
By using Pythagoras theorem, we can say that,
$ {{\left( hypotenuse \right)}^{2}}={{\left( height \right)}^{2}}+{{\left( base \right)}^{2}} $
$ \Rightarrow O{{Q}^{2}}=O{{P}^{2}}+P{{Q}^{2}} $
$ \therefore P{{Q}^{2}}=O{{Q}^{2}}-O{{P}^{2}} $
$ PQ=\sqrt{O{{Q}^{2}}-O{{P}^{2}}}=\sqrt{{{12}^{2}}-{{5}^{2}}}=\sqrt{144-25} $
$ \therefore PQ=\sqrt{119}cm $
Hence we got the length of PQ as \[\sqrt{119}cm\].
Option D is the correct answer.
Note:
Read the question carefully to get the figure. If your figure is wrong then probably the answer you get will also be wrong. So be careful when you draw the figure. As the tangent meets the circle at point P, the angle will be \[{{90}^{\circ }}.\]
