Question

In the given fig, if AP=2cm, BQ = 3cm and RC = 4cm then calculate the perimeter of $\Delta ABC$

Answer 1

Hint:  First, proceeding for this, we must see the figure to get the idea of the tangents of the circle. Then, we know that two tangents drawn from an external point to the circle are equal in length. Then, we know that perimeter P of the triangle is given by the sum of the sides of the triangle as P=AB+BC+CA and by using it, we get the desired perimeter.

Complete solution:

In this question, we are supposed to find the perimeter of $\Delta ABC$ when AP=2cm, BQ = 3cm and RC = 4cm.

So, before proceeding for this, we must see the figure to get the idea of the tangents of the circle as:

Then, we know that two tangents drawn from an external point to the circle are equal in length.

So, by using the above statement, we get the following conditions as:

AR and AP can be considered as tangents drawn to the circle from point A which gives AR=AP.

BQ and BP can be considered as tangents drawn to the circle from point B which gives BQ=BP.

CR and CQ can be considered as tangents drawn to the circle from point C which gives CR=CQ.

So, we get the values from the figure and above mentioned condition as:

AR =AP = 2

BQ = BP = 3

CR =CQ = 4

Now, we know that perimeter P of the triangle is given by the sum of the sides of the triangle as:

P=AB+BC+CA

Now, by using the values of the above lengths in the above formula, we get:

AB=AP+PB

AC=AR+RC

BC=BQ+QC

Then, we get the perimeter P by substituting all the values as:

$P=AP+PB+AR+RC+BQ+QC$

$\Rightarrow P=2+3+2+4+3+4$

$\Rightarrow P=18$

Hence, we get the perimeter of the triangle as 18cm.

 

Note: We should know that the tangents drawn from the same point to a circle are equal. The general formula of perimeters that you should remember are: the perimeter of a rectangle is$2(a+b)$, the perimeter of a square is $4a$, the perimeter of a circle is $2\pi r$.