D and E are points on side AB and AC of ΔABC. DE is parallel to BC. If AD : DB = 2 : 3 and area of ΔABC is 100 sq cm, what is the area (in sq cm) of quadrilateral BDEC?
84
Since \(DE \parallel BC\), \(\triangle ADE \sim \triangle ABC\) with ratio \(\dfrac{AD}{AB}=\dfrac{2}{2+3}=\dfrac{2}{5}\).
By the area ratio of similar triangles:
\[\frac{[\triangle ADE]}{[\triangle ABC]}=\left(\frac{2}{5}\right)^2=\frac{4}{25}\]
\[[\triangle ADE]=100\times\tfrac{4}{25}=16 \text{ sq cm}\]
Therefore \([BDEC]=100-16=\mathbf{84}\) sq cm.
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