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I'm trying to prove that length of the line $AB$, parallel to both bases of a trapezoid, that cuts a trapezoid into two trapezoids of equal area is the Root Mean Square of the bases. In other words, if the length of the top base is $a$ and the length of the other base is $b$, then :

$AB=\sqrt{\frac{a^2 + b^2}{2}}$

I've been stuck on this for a couple of days. Can someone give me a hint? Please don't just state the proof. I've already tried using the areas of the two trapezoids to get the RMS.

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What does it mean for a "line" (line segment?) to be "the Root Mean Square"? Are you describing the length of such a line segment? In adjuring us not to "just state the proof", are you emphasizing that you want only a hint, or is there another implied request? – hardmath Jan 15 '14 at 3:21
Sorry, I'd like a hint. I'll amend the question to make it more clear. – Josh Infiesto Jan 15 '14 at 3:22
Let the length of $AB$ be $c$, let the height of the trapezoid be $h$, and let the height of the sub-trapezoid between bases $a$ and $c$ be $k$ (so that the height of the sub-trapezoid between bases $b$ and $c$ is $h-k$). Proportionality arguments show that $c$ and $h$ are related by $$c = a + \frac{k}{h} ( b - a )$$ (Sanity check: Try $k=0$ and $k=h$.) Use this relation to write $k$ and $h-k$, and then also the areas of the two sub-trapezoids, in terms of $a$, $b$, $c$, and $h$. Setting the areas equal should give the relation you seek. – Blue Jan 15 '14 at 5:11
Use the fact that surface areas of two subtrapezoids are equal.On the other hand the sum of their surface areas is equal to the total surface area.From these two you can obtain the length of the required segment. – p.s Jan 15 '14 at 5:27
P.s. I mentioned I tried that approach. I couldn't get anywhere with it. Blue, can you expand on the proportionality arguments? I tried using similar figures to get proportions, but I couldn't show any helpful ones. – Josh Infiesto Jan 15 '14 at 7:37
up vote 2 down vote accepted

Converting comments to an answer, as requested ...

Suppose that $a$ is the shorter base. Draw a line through an endpoint of the $a$-base, parallel to the opposite lateral side; this cuts the trapezoid into a parallelogram (with upper and lower bases $a$) and a triangle (with (lower) base $b-a$). The drawn line separates your "middle base" $AB$, say of length $c$, into a part within the parallelogram and a part within the triangle; the first part has length $a$, while the second part has, by similar triangle proportions, length $\frac{k}{h}(b-a)$, where $h$ is the height of the triangle (and the original trapezoid) and $k$ is the height of the sub-triangle (and the height of the sub-trapezoid between bases $a$ and $c$). Thus, $$c = a + \frac{k}{h}(b-a)$$ Use this relation to write $k$ and $h-k$ (the heights of the two sub-trapezoids) in terms of $a$, $b$, $c$, and $h$. Then write the areas of the sub-trapezoids in terms of those parameters. Setting the areas equal gives the relation you seek.

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+10 !!! Very crafty that parallel line $\ddot \smile$ – K. Rmth Jan 17 '14 at 18:15

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