In the figure,$XY\parallel QR$and $\dfrac{PX}{XQ}=\dfrac{PY}{YR}=\dfrac{1}{2}$,find $XY:QR$

Hint:
From the given we can find the value of $\dfrac{PR}{PY}$ , Then compare the triangle $\Delta PXY\And \Delta PQR$, we get the common angle and corresponding angles. So we can say that they are equiangular. By $AAA$ similarity their corresponding sides are proportional. If we take the corresponding sides as proportional we get the $XY:QR$.

Complete step by step solution:
Given that,
$\Rightarrow \dfrac{PX}{XQ}=\dfrac{PY}{YR}=\dfrac{1}{2}$
From that we can find $\dfrac{PR}{PY}$
Taking reciprocals for $\dfrac{PY}{YR}$ we get,
$\Rightarrow \dfrac{YR}{PY}=\dfrac{2}{1}=2$
So adding the$PY+YR$, we get $PR$
$\Rightarrow \dfrac{PY+YR}{PY}=\dfrac{1+2}{1}$
$\Rightarrow \dfrac{PR}{PY}=\dfrac{3}{1}=3$
Then compare$\Delta PXY\And \Delta PQR$, we get
Common angle is $\angle P$
Since $XY\parallel QR$and $PQ$ is transversal then, Transversal means that the line passes through two line in the same plane at two distinct points,
$\Rightarrow \angle PXY=\angle PQR$ are the corresponding angles,
Since $XY\parallel QR$and $PR$ is transversal then,
$\Rightarrow \angle PYX=\angle PRQ$ are the corresponding angles, From that we get the angles
$\Rightarrow \angle PXY=\angle Q$
$\Rightarrow \angle PYX=\angle R$
$\Delta PXY\And \Delta PQR$are equiangular.
Equiangular means whose vertex angles are equal.
By $AAA$ similarity, means that two triangles have their corresponding angles equal if and only if their corresponding sides are proportional.
Now we have the corresponding angles are equal so their corresponding sides are proportional,
$\Rightarrow \dfrac{PX}{PQ}=\dfrac{XY}{QR}=\dfrac{PY}{PR}$ , From this we say that
$\Rightarrow \dfrac{XY}{QR}=\dfrac{PY}{PR}$ , we have $\dfrac{PR}{PY}=3$
So the reciprocals of $\dfrac{PR}{PY}=3$ is $\dfrac{PY}{PR}=\dfrac{1}{3}$
$\Rightarrow \dfrac{XY}{QR}=\dfrac{1}{3}$

$\Rightarrow XY:QR=1:3$

Note:
To prove $AAA$ similarity we have the statement that if in two triangles , the corresponding angles are equal, that is if the two triangles are equiangular, then the triangles are similar.