the Garden problem

[prog_tom]

There's a garden, using rectangular coord:

0,0

0,5

12,5


. Obviously, a right triangle.




Q:

#1: Where do I draw a horizontal line so that it can divide the triangle into to 2 equal sections. (y=?)

#2. Prove both section's area are equal after drawing the line.

#3. Where else can I draw a horizontal or vertical line to divide this into 2 equal areas?

[nishantp]

Look man, do your homework. Look in a text book. Or better yet, look in the INTERNET. You'll get an answer a lot faster...


To split a triangle by area, you find the midpoint of one side, and then connect it with the opposite vertex. You should have 2 triangles with the same area. Again, I'm not positive, so look it up.

[sql_lall]

Actually no, midpoints don't give you the answer if ur only allowed horizontal and vertical lines.

Anyway, here's the 'thinking' process:

1) You know you want to place a horizontal line.

2) You know how to work out the area of a triangle if you know its base and its height.

3) You know the area of the initial triangle

4) You know that after you place the horizontal line, somewhere between the top & bottom of the triangle, that your triangle will be split into two areas, the 'below-the-line' quadrilateral and the 'above-the-line' triangle

5) You know that the areas of these two are the same

6) You know that the sum of the areas is the area of the big triangle

7) => What do you know about what the area of the 'above-the-line' triangle has to be? (Try calling this area x)
Hint: you can relate x to the area of the whole triangle, which is constant.

8) Now, you know the base of this above-the-line triangle in terms of where you placed it (use similar triangles)

9) You also know the altitude

10) => You know the area of the above-the-line triangle in terms of where the horizontal line is.

11) => You have an equation relating where the horizontal line is to x, which is related to a constant.

12) => You have an equation with where the horzontal line is, = some other constant, which tells you where to put it.


Often, these "What do I know" type thought processes are extremely useful. If you are given lots of data in the question, then you are almost guarenteed that you'll use it in one of the initial "I know" statments. (Like, in this case, the area of the whole triangle, which you are given)