Question

From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA = 14cm, find the perimeter of \[\Delta PCD\] .

Answer 1

Hint: In order to solve this problem we need to draw the diagram and use the concept that two tangents on the circle from an external point are equal in length. We know the perimeter of a triangle is the sum of all the three sides of a triangle. So we need to find the value of each side and adding them all we get the required answer.

Complete step-by-step answer:

Let’s draw a diagram using the given data,

Given PA = 14 cm

We need to find the perimeter of \[\Delta PCD\]

We need to find the length of sides CD, PD and PC.

We know PA = PB = 14 cm.

Because two tangents on the circle from an external point are equal in length.

In the diagram we can see that there are two tangents from C and tangents from B.

So, we have DB = DE and CA=CE - (1)

From the diagram we have \[CD = CE + ED\]

 \[ \Rightarrow CD = CA + DB\] - (2)

This is because from (1).

Also \[PA = PC + CA\] and \[PB = PD + DB\] . - (3)

Now we know that perimeter of \[\Delta PCD\] is

 \[PC + CD + PD\]

 \[ = PC + CA + DB + PD\] (Because from (2)).

 \[ = PB + PB\] (From (3)).

Hence the perimeter of \[\Delta PCD\] is \[ \Rightarrow PC + CD + PD = PB + PB\] .

We know PA = PB = 14 cm.

Hence the perimeter of \[\Delta PCD = 14 + 14 = 28\;cm\] .

So, the correct answer is “28 cm”.

Note: The important concept that we used in this is two tangents on the circle from an external point are equal in length. We showed the perimeter of the triangle is equal to the sum of the two tangents drawn from the external point. So if they give the same kind of problem with different lengths of tangent we can find the perimeter directly by adding them.