How many different trapezoids are there in the figure given that all the horizontal line segments are parallel and no other segments are parallel?

Question

                                                          —————————————

                                                        ⁄      ⁄      \            \          \

                                                      ⁄—— ⁄ —  —\ — —  —\ — — —\

                                                    ⁄        ⁄           \          \              \

                                                  ⁄         ⁄             \         \                \  

                                                ⁄ —  — ⁄ — —   —  \ —   —\ —  —   —   \
                                              ⁄           ⁄                 \         \                   \ 
                                            ⁄ —   —   ⁄  —   —     —\  —— \  —    —      — \
                                          ⁄             ⁄                     \        \                      \ 

                                         ⁄             ⁄                        \       \                       \

                                       ————————————————————————————

 

*** All the lines I have drawn with the underscores and dashes, please connect them to make lines following the similar line pattern. I tired to do my best to make the trapezoid look like it does on my paper. The third horizontal line is supposed to be more appropriately spaced. 

 

Question: Use the Identify Subgoals strategy while trying to find: How many different trapezoids are there in the above figure given that all the horizontal line segments are parallel and no other segments are parallel?

 

The Identify Subgoals strategy is where this problem can best be solved by first identifying one or more intermediate conditions or "subgoals" whose results may lead to a solution to the original problem.

 

    ––I don't know how to make a subgoal or what it should be. Please help me identify a subgoal.

 

PLEASE HELP! THANK YOU!!! 

 

Answer 1

Hi Sarah,

 

I think what your teacher wants you to see is that the problem can be broken down into bite size pieces.

 

As best that I can tell from your diagram, it consists of four layers.  If we just look at the first layer and count how many trapezoids are possible it is a starting point for the entire problem.

 

so your first layer would look like 

     ———————————
   ⁄           ⁄       \            \           \
⁄———⁄———\———\———\

 

If we label this as having a top an bottom and the connecting sides from left to right are 1,2,3,4 & 5

It has four spaces or potential trapezoid bodies as there are five sides.

 

The top and bottom define 2 of the sides for all of these trapezoids, so I will just list the side pair numbers

 

Trapezoids 1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, 4-5, So this creates 10 different possibilities.

 

Now if we looked at the left most column of potential trazedoids, it is also 4 has spaces as there are 5 potential tops and bottoms.  So as in the horizontal model above this leftmost "column" of trapezoids has 10 possibilities.

 

If we then simply multiply 10 horizontal possibilities by 10 vertical possibilities we get that there are in fact 100

possible trapezoids that can be created.

 

I hope this helps.  If you have any more questions about it please ask.