Question

In the below figure, if $PS = 14$cm, the value of the $'\tan a'$ is equal to:

(A) $\dfrac{4}{3}$
(B) $\dfrac{{14}}{3}$
(C) $\dfrac{5}{3}$
(D) $\dfrac{{13}}{3}$

Answer 1

Hint: First find the length ST using the given data and then apply the Pythagoras theorem to find the length of $PQ$, which gives then the length of $RT$, after that apply the trigonometric ratio in the triangle to find the value of $\tan a$.

Complete step-by-step answer:
Details given in the figure are:
$PR = 13$cm, $QR = 5$cm, and $PS = 14$cm
Then we can find the length of $ST$ using the given lengths as:
$ST = PS - TP$
It can also be seen that $RQ = TP = 5$cm
Substitute the values $PS = 14$cm and $TP = 5$cm into the equation, so we have
$ST = 14 - 5$
$ST = 9$cm
Now, apply the Pythagoras theorem in a triangle $PQR$ and according to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the square of the base and the perpendicular.
$P{R^2} = P{Q^2} + Q{R^2}$
Substitute the values $PR = 13$cm and $QR = 5$cm into the equation:
${\left( {13} \right)^2} = P{Q^2} + {\left( 5 \right)^2}$
Solve the equation of the length of $PQ$.
\[PQ = \sqrt {{{\left( {13} \right)}^2} - {{\left( 5 \right)}^2}} \]
\[PQ = \sqrt {169 - 25} \]
\[PQ = \sqrt {144} \]
\[PQ = 12\]
So, we have the length $PQ = 12$cm.
It can be seen from the figure that $PQ = RT$, so we can conclude that:
$RT = 12$cm
Now, for the triangle $RST$, we have
$ST = 9$cm and $RT = 12$cm
Use the trigonometric ratio in the $RST$, so we have the $\tan $ of the angle $'a'$ is the ratio of the perpendicular and the base. That is,
$\tan a = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$
For the angle$'a'$, the perpendicular is $RT$ and the base is $ST$ whose values are:
$ST = 9$cm and $RT = 12$cm
So, substitute these values to get the trigonometric ratio. Now, we have
$\tan a = \dfrac{{RT}}{{ST}}$
$\tan a = \dfrac{{12}}{9}$
$\tan a = \dfrac{4}{3}$
Therefore, the obtained value of $\tan a$ is $\dfrac{4}{3}$.

Note:
In a right-angled triangle $RST$, the tan of the angle “a” is the ratio of the perpendicular and base. The perpendicular is the side which is just opposite to the angle, the base is the adjacent side to the angle and the rest side is the hypotenuse of the triangle.