3420.31 – The Trapezoid and the Triangle

The triangles $\triangle RVW$ and $\triangle RTS$ are similar, therefore the bases are in the ratio $\dfrac{5}{11}$. The bases themselves will be, say, $5k$ and $11k$ respectively, where $k$ is a constant.

The heights of the two triangles are in the same ratio, say, $5m$ and $11m$. Therefore the ratio of the areas ($\text{area} = \text{base} \times \text{height} \div 2)$ of the triangles is

$$\dfrac{\dfrac{5k \times 5h}{2}}{\ \dfrac{11k \times11h}{2}\ } = \dfrac{25}{121}$$

Now if the area of $\triangle RVW$ is $25z$, then the area of the trapezoid $STVW$ is $121z - 25z = 96z$ and the ratio of these areas is,

$$\dfrac{25z}{96z} = \dfrac{25}{96}$$