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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:

Attachment 23964

(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!

Re: Area of a triangle Question

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

Re: Area of a triangle Question

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

Re: Area of a triangle Question

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

Re: Area of a triangle Question

Re: Area of a triangle Question

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!

Re: Area of a triangle Question

Quote:

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....