Question

In the figure PA and PB are two tangents drawn from an external point P to a circle with centre C of radius 4 cm. If \[PA\bot PB\] then the length of each tangent is:

(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm

Answer 1

Hint: We solve this problem by using the properties of tangents.
We have the condition that the radius of the circle is perpendicular to the tangent.
We also have the condition that the tangents drawn to the circle from the external point are equal
By using the above two conditions we find the value of the required length.

Complete step by step answer:
We are given that the radius of the circle as 4 cm
By using the above condition to the given figure we get
\[\Rightarrow CA=CB=4cm\]
We know that the condition that the tangents drawn to the circle from the external point are equal

By using the above condition to the given figure we get
\[\Rightarrow PA=PB\]
We are given that \[PA\bot PB\]
We know that the condition that the radius of the circle is perpendicular to the tangent.
By using the above condition we given figure then we get \[CA\bot PA\] and \[CB\bot PB\]
Now, let us consider the quadrilateral CAPB
We have two conditions for the quadrilateral CAPB as
(1) All angles are equal to \[{{90}^{\circ }}\]
(2) Adjacent sides are equal \[\left( PA=PB\And CA=CB \right)\]
By using the above two conditions we can say that the only possibility for such type of quadrilateral is square.
So, we can say that CAPB is a square.
We know that in a square all the sides are equal.
By using the above condition to given information we get
\[\Rightarrow CA=CB=PA=PB=4cm\]
Therefore, we can conclude that the length of each tangent is 4 cm
So, option (b) is the correct answer.

Note:
 We need to note that if all the angles of a quadrilateral are equal to \[{{90}^{\circ }}\] then there will be two possibilities for the quadrilateral as square or rectangle.
But in a rectangle, the opposite sides will be equal and in a square all sides are equal.
So, if there is a possibility that adjacent sides are equal then the only possibility is square.