3420.31 – The Trapezoid and the Triangle

The triangles RVW and RTS are similar, therefore the bases are in the ratio $\frac{5}{11}.$ The bases themselves will be, say, $5k$ and $11k$ respectively, where $k$ is a constant. The altitudes of the two triangles are in the same ratio, say, $5m$ and $11m$. Therefore the ratio of the areas of the triangles is

$\frac{\frac{1}{2} 5k \cdot 5m}{\frac{1}{2} 11k \cdot 11m} = \frac{25}{121}.$

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

$\frac{25z}{96z} = \frac{25}{96}.$