3420.31 – The Trapezoid and the Triangle

Find the ratio of the area of the triangle RVW to the area of the trapezoid STVW, where VW is parallel to TS.



The triangles RVW and RTS are similar, therefore the bases are in the ratio 511. \frac{5}{11}. The bases themselves will be, say, 5k5k and 11k11k respectively, where kk is a constant. The altitudes of the two triangles are in the same ratio, say, 5m5m and 11m11m. Therefore the ratio of the areas of the triangles is

125k5m1211k11m=25121. \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, 25z25z, then the area of the trapezoid STVW is 121z25z=96z121z - 25z = 96z and the ratio of these areas is,

25z96z=2596. \frac{25z}{96z} = \frac{25}{96}.