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I have a right trapezoid as follows;

enter image description here

We have $h$, $b$ and $a$. For any $n$, I need to divide total area of trapezoid into equal parts. I have to find a general formulation for the length of $p$ points for my study. Although I try to figure out from starting from $n=2$, $n=3$,... I couldn't able to generalize the formulation. Is there any general formulation for this?

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thanks for editing, I couldn't able to put images with my current reputation :) –  knightoi Dec 5 '11 at 23:51
    
Do you mean that you have a solution for $n=2$ and $n=3$ and just need to generalize it to arbitrary $n$? If so, please show your existing work. –  Henning Makholm Dec 6 '11 at 0:06

1 Answer 1

Complete the rectangle:

   A ______________________________________ B  
    |   .   |  x   |     |                 |  
    |       °  .   |  x' |   Triangle      |  
    |       |      °  .  |                 |  
    |       |      |     |   .             |  
 b  |       |      |     |          .      |  
    |       |      |     |                 |C  
    |       |  y   |  y' |                 |  
    |       |      |     |                 |  a  
    |       |      |     |                 |
   E|_______|______|_____|_________________|D  
                       h

Note that the sections of the trapezoid have equal areas if and only if the corresponding sections of the triangle have equal areas. For example, area $y'$ equals area $y$ if and only if area $x'$ equals area $x$. Let $x_1,x_2,\dots,x_n$ be the distances of the vertical lines from the the line $\overline{AE}$, in order from left to right, so that $x_n=h$. Let $m=\frac{b-a}h$. Then the areas of the first three sections of the triangle are

$$\begin{align*} &\frac12mx_1^2,\\ &\frac12mx_2^2-\frac12mx_1^2,\text{ and}\\ &\frac12mx_3^2-\frac12mx_2^2-\frac12mx_1^2, \end{align*}$$

and the pattern should be clear.

We need to ensure that these are all equal. We might as well multiply them all by $2/m$ and instead try to ensure that

$$x_1^2=x_2^2-x_1^2=x_3^2-x_1^2-x_2^2=\dots=x_n^2-x_1^2-\cdots-x_{n-1}^2,$$

i.e., that

$$\begin{align*} x_2^2&=2x_1^2,\\ x_3^2&=x_1^2+x_2^2=(1+2)x_1^2=3x_1^2,\\ x_4^2&=x_1^2+x_2^2+x_3^2=(1+2+3)x_1^2=6x_1^2, \end{align*}$$

and so on. With that much of a start you should be able to attack these questions successfully:

  1. What is $x_k^2$ in terms of $x_1^2$?
  2. In particular, what is $x_n^2$ in terms of $x_1^2$?
  3. What is $x_1$ in terms of $h$?
  4. Do $a$ and $b$ have any effect on the result?
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