Question

In $\vartriangle PQR$, $\angle Q: \angle R:\angle P = 1:2:1$. If $PR = 2\sqrt 3 $ then, PQ=____.
A. $2\sqrt 3 $
B. $2\sqrt 6 $
C. $12$
D. $4\sqrt 6 $

Answer 1

Hint:
In this question, we need to determine the length of the side PQ of the triangle PQR such that the ratio of the interior angles of the triangle is given as $\angle Q:\angle R:\angle P = 1:2:1$. For this, we will use the property of the triangle, which states that the sum of the interior angles of the triangle is equal to 180 degrees to evaluate the angles and then, apply the Pythagoras theorem to evaluate the length of the sides.

Complete step by step solution:
In the triangle PQR, the interior angles of the triangle are given in the ratio of $m\angle Q:m\angle R:m\angle P = 1:2:1$.

Let the common multiplier be $x$. So, the interior angles of the triangle PQR are $x,2x{\text{ and }}x$ for $\angle Q,\angle R{\text{ and }}\angle P$ respectively.
The summation of the interior angles of a triangle equals to 180 degrees. So, $\angle Q + \angle R + \angle P = {180^0} - - - - (i)$
Substituting the expression for the angles in the equation (i) we get,
$
  \angle Q + \angle R + \angle P = {180^0} \\
  x + 2x + x = {180^0} \\
  4x = {180^0} \\
  x = \dfrac{{{{180}^0}}}{4} \\
   = {45^0} \\
 $
As, $x = {45^0}$ so, the interior angles in the triangle PQR is
$
  \angle Q = x = {45^0} \\
  \angle R = 2x = 2 \times {45^0} = {90^0} \\
  \angle P = x = {45^0} \\
 $
From the calculation, we can say that the triangle PQR is a right angle triangle with angle R at 90 degrees.

Following the property of the triangles, if the two interior angles of the triangle are equal then, their corresponding opposite sides are also equal and will form an isosceles triangle.
So, here $\angle Q = \angle P = {45^0}$ and hence, $PR = RQ$.
According to the question, $PR = 2\sqrt 3 $ so, $PR = RQ = 2\sqrt 3 $
Now, following the Pythagoras theorem in the right-angle isosceles triangle PQR in which PQ is the hypotenuse of the triangle.
$
  {\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Height} \right)^2} \\
  {(PQ)^2} = {(PR)^2} + {(RQ)^2} - - - - (ii) \\
 $
Substituting the values of PR and RQ as $PR = RQ = 2\sqrt 3 $ in the equation (ii), we get
$
  {(PQ)^2} = {(PR)^2} + {(RQ)^2} \\
   = {\left( {2\sqrt 3 } \right)^2} + {\left( {2\sqrt 3 } \right)^2} \\
   = 12 + 12 \\
   = 24 - - - - (iii) \\
 $
Applying square root to both sides of the equation (iii), we get
$
  {\left( {PQ} \right)^2} = 24 \\
  PQ = \sqrt {24} \\
   = 2\sqrt 6 \\
 $
Hence, $PQ = 2\sqrt 6 $ in the triangle PQR.

Option B is correct.

Note:
Students must note here that the common multiplier has been taken in order to evaluate the absolute values of the interior angles of the triangles as here, in this question, the ratio of the interior angles is given.