Question

The length of the tangent to a circle from a point $P$, which is $25$ cm away from the centre is $24$ cm. What is the radius of the circle?

Answer 1

We are given that the length of the tangent to a circle from a point $P$, which is $25$ cm away from the centre is $24$ cm. We can say that point $P$ is an external point. So, use the property and solve it by using Pythagoras theorem.

Complete step-by-step answer:
Draw a circle and let $P$ be the point such that $OP=25$ cm and also, $TP$ is a tangent which is $24$cm.
We can see in the figure that $OT$ is the radius.

We know tangent drawn from an external point is perpendicular to radius at the point of contact.
Now in figure as $OT$ is radius and $TP$ is a tangent, we can say that $OT$ is perpendicular to $TP$.
$OT\bot TP$
Now $\Delta OTP$ is a right angled triangle.
By using Pythagoras theorem we get,
${{(OP)}^{2}}={{(OT)}^{2}}+{{(TP)}^{2}}$
So, arranging in proper manner we get,
$(OT)=\sqrt{{{(OP)}^{2}}-{{(TP)}^{2}}}$
We know $OP=25$cm and $TP=24$cm.
Substituting $OP=25$ and $TP=24$ in above we get,
$(OT)=\sqrt{{{(25)}^{2}}-{{(24)}^{2}}}$
$(OT)=\sqrt{625-576}$
Now simplifying we get,
$(OT)=\sqrt{49}$
We get, $(OT)=7$cm.
Therefore, the length of radius is $7$cm.
Additional information:
Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle $90{}^\circ$.
A circle is the locus of points which moves in a plane such that its distance from a fixed point is always constant. The fixed point is called the ‘centre’ while the fixed distance is called the ‘radius’.

Note: We have used a basic property which is important in this problem which is tangent drawn from an external point is perpendicular to radius at the point of contact. Also, we must able to visualize the figure correctly