In the diagram above QRST, QR: TS = 2:3 find the ratio of the area of triangle PQR to the area of the trapezium QRST.

Steps to Solve

1. Identify the given ratio
   We know from the problem that the ratio of segments $QR$ to $TS$ is given as $QR:TS = 2:3$. This implies that if the length of $QR$ is $2x$, then the length of $TS$ is $3x$ for some value of $x$.

2. Calculate the area of triangle PQR
   The area of triangle PQR can be computed using the formula for the area of a triangle:

$$ \text{Area}_{\triangle PQR} = \frac{1}{2} \times \text{base} \times \text{height} $$

In this case, the base can be taken as $QR$ and the height will be the segment $PQ$. Denote the height as $h$, so:

$$ \text{Area}_{\triangle PQR} = \frac{1}{2} \times 2x \times h = xh $$

3. Calculate the area of trapezium QRST
   The area of a trapezium can be calculated with the formula:

$$ \text{Area}_{\text{trapezium}} = \frac{1}{2} \times (b_1 + b_2) \times h $$

In trapezium QRST, let $b_1 = QR = 2x$ and $b_2 = TS = 3x$. The height is the same height as triangle PQR ($h$). Therefore:

$$ \text{Area}_{\text{trapezium QRST}} = \frac{1}{2} \times (2x + 3x) \times h = \frac{5x}{2} \times h $$

4. Find the ratio of the areas
   Now we can find the ratio of the area of triangle PQR to the area of trapezium QRST:

$$ \text{Ratio} = \frac{\text{Area}{\triangle PQR}}{\text{Area}{\text{trapezium}}} = \frac{xh}{\frac{5x}{2}h} $$

This simplifies to:

$$ \text{Ratio} = \frac{xh \cdot 2}{5xh} = \frac{2}{5} $$

5. Express the final ratio
   The final ratio is therefore:

$$ \text{Ratio} = 2:5 $$