find the equation of line that divides the trapezoid into two equivalent regions

Question

Given that $A(1,0)$, $B( 3,0)$, $C(3,5)$, $D(1,4)$ are the vertices of trapezoid $ABCD$, find the equation of a line that divides the trapezoid into two equivalent regions?

Answer 1

Half of the area of the trapezoid is given by $\dfrac {1}{2} \times \dfrac {(4 + 5)2}{2}$.

After joining BD, we found that $[\triangle ABD] \lt [\triangle BCD]$. Therefore, we need to cut some from $\triangle BCD$ and attach the cut-off to $\triangle ABD$. Then, the target line need to go through D and P(3, k) as well; where P should be a point somewhere between BC.

Then, [Trap ABPD] = $\dfrac {(k + 4)(2)}{2} = \dfrac {1}{2} \times \dfrac {(4 + 5)2}{2}$.

From that, we get k = 0.5.

Finally use two-point form to write the equation of DP, one of the required lines.