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 lengths and ratios We know that the ratio of segments QR and TS is given as $QR:TS = 2.3$. We can denote the lengths of QR as $2.3x$ and TS as $x$ for simplification.

2. Determine the heights Since both triangle PQR and trapezium QRST share the same height from point P down to line QS, we will refer to this height as $h$.

3. Calculate the area of triangle PQR The formula for the area of a triangle is given by: $$ \text{Area of } \triangle PQR = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times QR \times h = \frac{1}{2} \times 2.3x \times h = 1.15xh $$

4. Calculate the area of trapezium QRST The formula for the area of a trapezium is: $$ \text{Area of trapezium QRST} = \frac{1}{2} \times (b_1 + b_2) \times h $$ Here, $b_1 = QR = 2.3x$ and $b_2 = TS = x$. Therefore, $$ \text{Area of trapezium QRST} = \frac{1}{2} \times (2.3x + x) \times h = \frac{1}{2} \times 3.3x \times h = 1.65xh $$

5. Find the ratio of areas To find the ratio of the area of triangle PQR to the area of trapezium QRST, we set up the ratio: $$ \text{Ratio} = \frac{\text{Area of } \triangle PQR}{\text{Area of trapezium QRST}} = \frac{1.15xh}{1.65xh} $$ The $xh$ cancels out, giving: $$ \text{Ratio} = \frac{1.15}{1.65} $$

6. Simplify the ratio To simplify $\frac{1.15}{1.65}$, we convert it into a simpler fraction: $$ \frac{1.15}{1.65} = \frac{115}{165} $$ To simplify, we find the GCD (greatest common divisor) of 115 and 165, which is 5. Dividing both by 5 gives us: $$ \text{Simplified Ratio} = \frac{23}{33} $$