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One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$.

I used the given information to set up a ratio and was able to calculate that the bases of the trapezoid were 75 and 175. I am not sure how to proceed to find the length of the segment dividing into equal areas though. Anyone have any ideas?

Dude156
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1 Answers1

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Let $h$ be the height of the trapezoid (it does not really matter). If you draw a line parallel to the bases and it has length $75+y$, then the distance $d$ between this line and the small base must satisfy $\frac{y}{d} = \frac{175-75}{h}$, i.e. $d=yh/100$. The area of the two small trapezoids are then $\frac{1}{2} (150 + y) yh/100$ and $\frac{1}{2} (75+y + 175) h(1-y/100)$. Cancel the $h$ and solve for $y$.

angryavian
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