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# help geometry

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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?

Mar 11, 2022

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What is the area of triangle BEC?

Hello Guest!

The perpendicular through C intersects the line through AB at point F.

Then

$$f(x)=\frac{80}{28}x=\frac{20}{7}x\\ g(x)=-\frac{80}{7}x+80\\ f(x)=g(x)\\ \frac{20}{7}x=-\frac{80}{7}x+80\\ \frac{100}{7}x=80$$

$$x_E=5.6\\ y_E=16$$

$$A_{BEC}=A_{ACF}-A_{BCF}-A_{ABE}$$

$$A_{ACF}=\frac{28\cdot 80}{2}=1120\\ A_{BCF}=\frac{80(28-7)}{2}=840\\ A_{ABE}=\frac{7\cdot y_E}{2}=\frac{7\cdot 16}{2}=56$$

$$A_{BEC}=1120-840-56\\ \color{blue}A_{BEC}=224$$

!

Mar 12, 2022