ABCD is a right trapezoid with AB parallel to CD and ∠BAD=∠ADC=90∘. The diagonals AC and BD intersect at E. If AB=7, DC = 28 and AD = 80, what is the area of triangle BEC?
+15000 What is the area of triangle BEC?
Hello Guest!
The perpendicular through C intersects the line through AB at point F.
Then
\(f(x)=\frac{80}{28}x=\frac{20}{7}x\\ g(x)=-\frac{80}{7}x+80\\ f(x)=g(x)\\ \frac{20}{7}x=-\frac{80}{7}x+80\\ \frac{100}{7}x=80\)
\(x_E=5.6\\ y_E=16\)
\(A_{BEC}=A_{ACF}-A_{BCF}-A_{ABE}\)
\(A_{ACF}=\frac{28\cdot 80}{2}=1120\\ A_{BCF}=\frac{80(28-7)}{2}=840\\ A_{ABE}=\frac{7\cdot y_E}{2}=\frac{7\cdot 16}{2}=56\)
\(A_{BEC}=1120-840-56\\ \color{blue}A_{BEC}=224\)
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