In the diagram above QRTS, 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 segments and ratio The given ratio is ( QR:TS = 2:3 ). This means that if we denote ( QR = 2x ) and ( TS = 3x ), we can express the lengths in terms of a variable ( x ).

2. Calculate the area of triangle PQR The area of triangle PQR can be calculated using the formula: $$ \text{Area}{\triangle} = \frac{1}{2} \times \text{base} \times \text{height} $$ Here, ( QR ) is the base of the triangle and the height can be denoted by ( h ). Thus, $$ \text{Area}{\triangle PQR} = \frac{1}{2} \times QR \times h = \frac{1}{2} \times 2x \times h = xh $$

3. Calculate the area of trapezium QRST The area of trapezium QRST can be calculated using the formula: $$ \text{Area}{\text{trapezium}} = \frac{1}{2} \times (b_1 + b_2) \times h $$ where ( b_1 = QR = 2x ) and ( b_2 = TS = 3x ). Hence, $$ \text{Area}{QRST} = \frac{1}{2} \times (2x + 3x) \times h = \frac{1}{2} \times 5x \times h = \frac{5xh}{2} $$

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: $$ \text{Ratio} = \frac{\text{Area}{\triangle PQR}}{\text{Area}{QRST}} = \frac{xh}{\frac{5xh}{2}} = \frac{xh \times 2}{5xh} = \frac{2}{5} $$