Question

PQR is a right angled triangle in which $\angle R = 90^\circ $ . If $RS \bot PQ,PR = 3\,cm$ and $RQ = 4\,cm$, then what is the value of RS in cm?
A. $\dfrac{{12}}{5}$
B. $\dfrac{{36}}{5}$
C. $5$
D. $2.5$

Answer 1

Hint: In this question we have been given that the triangle is aright angled triangle. So we will use the Pythagoras theorem to solve this question. After that we will try to find the similarity between the triangles since it is given that RS is perpendicular to PQ, so we will have two triangles. We know that the Pythagoras theorem says that
$h = \sqrt {{p^2} + {b^2}} $ .

Complete answer:
According to the question PQR is aright angled triangle and we have $PR = 3\,cm$ and $RQ = 4\,cm$ .
Now let us draw the diagram of the triangle according to the question:

Here we have the hypotenuse of the triangle i.e. $PQ = h = ?$
And we have the perpendicular and the base i.e.
$PR = p = 3,RQ = b = 4$ .
By using the Pythagoras theorem we will calculate the hypotenuse, so we will substitute the values in the formula and we have:
$ \Rightarrow h = \sqrt {{3^2} + {4^2}} $
ON simplifying we have:
$ \Rightarrow h = \sqrt {25} $
It gives us the hypotenuse of the triangle i.e.
$PQ = 5\,cm$
Now we know that RS is perpendicular to the hypotenuse i.e. PQ , so we can say that the triangles are similar i.e.
$\Delta PRS \approx \Delta SRQ$ .
Again we know that if two triangles are similar then we can write the sides as:
$ \Rightarrow \dfrac{{PQ}}{{PR}} = \dfrac{{QR}}{{RS}}$
By substituting the values in the equation, we have:
$ \Rightarrow \dfrac{5}{3} = \dfrac{4}{{RS}}$
On cross multiplying the values, we can write:
$ \Rightarrow RS = \dfrac{{4 \times 3}}{5}$
Therefore it gives us the value:
$RS = \dfrac{{12}}{5}$ .
Hence the correct option is (a) $\dfrac{{12}}{5}$ .

Therefore, the correct option is A

Note: We should note that both the triangles are similar by SAS criteria i.e. Side Angle Side criteria.
In $\Delta PRS$ and $\Delta RSQ$ , we have
$RS = RS$ (common side) ,
$PS = SQ$( Since RS is perpendicular to PQ)
And $\angle PSR = \angle QSR = 90^\circ $ .
Therefore both the triangles are similar by SAS criteria.