The length of the tangents from a point A to a circle of radius 3cm is 4cm. The distance (in cm) of A from the center of the circle is:

The length of the tangents from a point A to a circle of radius $3$cm is $4$cm. The distance (in cm) of A from the center of the circle is:

1. A
$\sqrt{7}$
2. B
$7$
3. C
$5$
4. D
$25$

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Solution:

Given a circle with radius 3cm and length of tangent 4 cm.
Drawing the required figure,
We use the fact that tangent is perpendicular to radius at the point of contact.
Using Pythagoras theorem in $\mathit{\Delta OBA}$,
${\mathit{AO}}^{2}={\mathit{OB}}^{2}+{\mathit{AB}}^{2}$
$⇒{\mathit{AO}}^{2}={3}^{2}+{4}^{2}$
$⇒\mathit{AO}=\sqrt{{3}^{2}+{4}^{2}}$
$⇒\mathit{AO}=5\mathit{cm}$ So, the distance of the point $A$ from the center of the circle is 5cm.
Therefore, option (3) is correct.

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