Ratio of Triangle Areas

This is to find the area of a triangle, when the area of another triangle is known. The two triangles have one concurrent angle, and the four lengths of the sides forming the angles are known. Or the ratios of corresponding sides are known.

This is based on the formula \[\text{triangle area }= \frac 1 2 \times a\times b\times \sin\gamma.\]

\(ABC\) is a triangle with a point \(D\) on the side \(AC\) and \(E\) on \(AB\) such that \(AE=3EB\) and \(DC=4AD.\)
Find the area of \(AED\) if the area of \(ABC\) is \(60.\)

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We have
\[\dfrac {\text{Area } \Delta AED}{\text{Area } \Delta AED}=\dfrac {\text{Area } \Delta AED}{60}=\dfrac 1{1+3}\times \dfrac 4{1+4}=\frac 1 5.\] Therefore, the area of \(\Delta AED\) is \(60\times \dfrac 1 4\times \dfrac 4 5=12.\) \(_\square\)