SOLUTION: Five lines parallel to the base of a triangle divide the other sides of the triangle each into 6 equal segments and the area into 6 distinct parts. If the area of the largest of th

Algebra ->  Volume -> SOLUTION: Five lines parallel to the base of a triangle divide the other sides of the triangle each into 6 equal segments and the area into 6 distinct parts. If the area of the largest of th      Log On


   

Question 833510: Five lines parallel to the base of a triangle divide the other sides of the triangle each into 6 equal segments and the area into 6 distinct parts. If the area of the largest of these parts is 66 cm squared then the area of the original triangle is:
I have drawn this out and played around with the lines but I do not see how I can find the area of any of the 6 sections
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Each section plus everything above that section base is a triangle.
They are all similar triangles with the side lengths in the proportion
%221+%3A+2+%3A+3+%3A+4+%3A+5+%3A+6%22.
The length of base and height are also in the same proportion, of course.
So, the areas are in the proportion
1%5E2%3A2%5E2%3A3%5E2%3A4%5E2%3A5%5E2%3A6%5E2=1%3A4%3A9%3A16%3A25%3A36
The area of the triangle made up of the top two sections is 4 times the area of the smallest (top) triangle.
The area of the triangle made up of the top three sections is 9 times the area of the smallest (top) triangle.
The area of the triangle made up of the top four sections is 16 times the area of the smallest (top) triangle.
The area of the triangle made up of the top five sections is 25 times the area of the smallest (top) triangle.
The original triangle's area is 36 times the area of the smallest (top) triangle.
Let's call the area of the smallest (top) triangle x square centimeters.
The area of the triangle made up of the top five sections is 25x square centimeters.
The original triangle's area is 36x square centimeters.
The difference, 36x-25x=11x , is the area, in square centimeters of the bottom section.
Our equation is 11x=66 --> x=66%2F11 --> x=6.
The area of the original triangle is
36x=36%2A6=highlight%28216%29 square centimeters.

A picture is worth a thousand words:
I added green lines to split the whole thing into 36 triangles that are congruent with (the same as) the smallest (top) triangle.
Do you see hoe there are 36 small triangles in all and 11 of them in the bottom section?