In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle$ADE$ is 20 square units, what is the area of trapezoid $ABCD$?
+26367 In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$.
If the area of triangle $ABE$ is 50 square units, and the area of triangle$ADE$ is 20 square units,
what is the area of trapezoid $ABCD$?
\(\text{Let Area $A_1=[ADE] = 20 $}\\ \text{Let Area $A_2=[ABE] = 50 $}\\ \text{Let Area $A_3=[DEC] $}\\ \text{Let Area $A_4=[BEC] $}\\ \text{Let Area $A_5=[ADC] $}\\ \text{Let Area $A_6=[BDC] $}\)
\(\begin{array}{|rcll|} \hline \dfrac{A_2}{A_1} &=& \dfrac{50}{20} \\ &=& \dfrac52 \\\\ \hline A_1 &=& \dfrac{DE\cdot AH}{2} \\\\ A_2 &=& \dfrac{BE\cdot AH}{2} \\\\ A_3 &=& \dfrac{DE\cdot CK}{2} \\\\ A_4 &=& \dfrac{BE\cdot CK}{2} \\ \hline \\ \dfrac{A_2}{A_1} &=& \dfrac{\dfrac{BE\cdot AH}{2}}{\dfrac{DE\cdot AH}{2}} \\\\ &=& \dfrac{BE\cdot AH}{DE\cdot AH} \\\\ \dfrac{A_2}{A_1} &=& \dfrac{BE}{DE} \quad & | \quad \dfrac{A_2}{A_1} = \dfrac52 \\\\ \mathbf{\dfrac52} & \mathbf{=}& \mathbf{\dfrac{BE}{DE}} \\\\ \hline \dfrac{A_4}{A_3} &=& \dfrac{\dfrac{BE\cdot CK}{2}}{\dfrac{DE\cdot CK}{2}} \\\\ &=& \dfrac{BE\cdot CK}{DE\cdot CK} \\\\ \dfrac{A_4}{A_3} &=& \dfrac{BE}{DE} \quad & | \quad \dfrac{BE}{DE} = \dfrac52 \\\\ \mathbf{\dfrac{A_4}{A_3}} &\mathbf{=}& \mathbf{\dfrac52} \qquad \text{or} \qquad \mathbf{A_3 = \dfrac{2}{5}\cdot A_4 } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline AM &=& BN \\ \hline \\ A_5 &=& \dfrac{DC\cdot AM}{2} \\\\ A_6 &=& \dfrac{DC\cdot BN}{2} \\ \hline \\ \dfrac{A_5}{A_6} &=& \dfrac{\dfrac{DC\cdot AM}{2}}{\dfrac{DC\cdot BN}{2}} \\\\ &=& \dfrac{DC\cdot AM}{DC\cdot BN} \quad & | \quad AM=BN \\\\ &=& \dfrac{DC\cdot BN}{DC\cdot BN} \\\\ \dfrac{A_5}{A_6} &=& 1 \\\\ \mathbf{ A_5 } & \mathbf{=} & \mathbf{A_6} \quad & | \quad A_5 = A_1+A_3,\quad A_6 =A_4+A_3 \\ A_1+A_3 &=& A_4+A_3 \quad & | \ -A_3\\ A_1 &=& A_4 \quad & | \quad A_1 = 20\\ \mathbf{20} & \mathbf{=}& \mathbf{ A_4 } \\ \hline \\ \mathbf{A_3} & \mathbf{=} & \mathbf{ \dfrac{2}{5}\cdot A_4 } \quad | \quad A_4 = 20 \\ A_3 &=& \dfrac{2}{5}\cdot 20 \\ \mathbf{A_3} & \mathbf{=} & \mathbf{ 8 } \\ \hline \end{array}\)
\(\text{The area of trapezoid $ABCD = A_1+A_2+A_3+A_4 = 20+50+8+20=\mathbf{98}$ }\)
