Thread: Area of a triangle Question

Hi,

I found this question and I could do with some help regarding the answer. It is:

Nine lines are drawn parallel to the base of a triangle to divide each of the other sides into 10 equal segments. If the area of the largest region is 38, find the area of the original triangle.

I believe that it must be ratios and something like:

(Sorry bad drawing!)

Ratio of sides =
Therefore, ratio of areas =




Therefore area =

Am I right? I feel that this is rather inadequate and too simple.

Thanks very much for any help!

To "see" how this works, start with a right triangle: 10 by 24 by 26 will make it easier...

Alternatively you could use the standard formula for the area of a triangle with sides and included angle

Area =

Call the sloping sides and say, then the area of the last strip will be

in which case so that the area of the whole triangle will be

A model triangle has an altitude of 10 units and a base of10 units. Its area = 50 sq units
Area of largest trapezoid section = 50 - 9*9/2 =9.5 sq units
Fractional area of above 9.5/50 = 0.19
Area of actual triangle* 0.19 =38
AAT = 200 sq units

Hello, BobtheBob!

Here's another approach . . .

Nine lines are drawn parallel to the base of a triangle to divide each of the other sides into 10 equal segments.
If the area of the largest region is 38, find the area of the original triangle.

Code:

    -         *
    :        *  *
    :       *     *
    :      *        *
    h     *           *
    :    *   (9/10)b    *
    :   * * * * * * * * * * - - -
    :  *                    *  (1/10)h
    - *  *  *  *  *  *  *  *  * -
      : - - - - - b - - - - - :

The base of the triangle is ; its height is

The bottom trapezoid has height and bases and

Its area is 38: .

Therefore: .

Thanks! So I was right with 200. However, all the solutions here are far better than mine in my opinion!

Thanks again for the help!

Originally Posted by BobtheBob

Nine lines are drawn parallel to the base of a triangle to divide each of the other sides into
10 equal segments. If area of the largest region is 38, find the area of the original triangle.

Givens are:
j = area largest region (38)
h = height of triangle (10)
n = number of regions (10)
Find:
k = area of triangle (?)

k = bh / 2
where b = 2jn(n - 2) / [2(h/n)(n^2 - 3n + 1) + h]
(b = base of triangle)

Substitute the givens and you'll get b = 40, hence k = 40(10) / 2 = 200

Now Bob, you can use that as a "general case" formula;
weird ones like j = 137, h = 23, n = 7 handled quickly!
That one will give you b = 44.903.... and k = 516.384....