Find the sides of right triangle, given only the area

happybeans said:
How to find the sides of the right triangle with the area of \(\displaystyle 9\sqrt{2}\)?
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There will be many different right triangles with any given area. Choose a value for b, and you can solve for h.

If this is part of a larger problem, please state the entire problem as given to you.
 
Dr.Peterson said:
There will be many different right triangles with any given area. Choose a value for b, and you can solve for h.

If this is part of a larger problem, please state the entire problem as given to you.
Click to expand...

I drew a picture, but it cannot be uploaded here, because it is too big. So the height height divides the right triangle in two right triangles with different areas.
 
happybeans said:
I drew a picture, but it cannot be uploaded here, because it is too big. So the height height divides the right triangle in two right triangles with different areas.
Click to expand...

The base and height I had in mind are the two legs! But it doesn't matter; however you look at it, there is not a unique solution, unless you have not yet told us the actual problem, and there is an additional constraint.

If you can't reduce the size of the picture (there are various free online sites you could use, if necessary), just describe it enough so that we can reproduce it. But also be sure to tell us the entire problem..
 
happybeans said:
I drew a picture, but it cannot be uploaded here, because it is too big. So the height height divides the right triangle in two right triangles with different areas.
Click to expand...
You are given a problem in which you are asked to find certain numbers. In most cases, if you are asked to find n related numbers, you must be know at least n mathematical relationships among those numbers.

I am guessing that there are nine related numbers.

Let a, b, and c, c being the length of the hypotenuse of the big triangle, and a and b being the lengths of the other two sides. Let x = area of the big triangle. What three mathematical relationships do you know with respect to a, b, c, and x?

Let h, j, and a be the lengths of the sides of one of the smaller triangles, with h being the length of the side common to both small triangles and j being the the length of the side lying on the hypotenuse of the larger triangle. Let y = the area of that triangle. What two mathematical relationships do you know with respect to h, j, a, and y?

Let h, k, and b be the lengths of the sides of the other smaller triangles, with h being the length of the side common to both small triangles and k being the length of the side lying on the hypotenuse of the larger triangle. Let z = the area of that triangle. What two mathematical relationships do you know with respect to h, p, b, and z?

Is there a relationship among, c, j, and k? If so. what is it?

Is there a relationship among x, y, and z? If so, what is it?

That makes up nine relationships so you may have enough information to solve the problem. However, it may not be enough (if some of the relationships are redundant). Is there any other relationship given?
 
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JeffM said:
You are given a problem in which you are asked to find certain numbers. In most cases, if you are asked to find n related numbers, you must be know at least n mathematical relationships among those numbers.

I am guessing that there are nine related numbers.

Let a, b, and c, c being the length of the hypotenuse of the big triangle, and a and b being the lengths of the other two sides. Let x = area of the big triangle. What three mathematical relationships do you know with respect to a, b, c, and x?

Let h, j, and a be the lengths of the sides of one of the smaller triangles, with h being the length of the side common to both small triangles and j being the the length of the side lying on the hypotenuse of the larger triangle. Let y = the area of that triangle. What two mathematical relationships do you know with respect to h, j, a, and y?

Let h, k, and b be the lengths of the sides of the other smaller triangles, with h being the length of the side common to both small triangles and k being the length of the side lying on the hypotenuse of the larger triangle. Let z = the area of that triangle. What two mathematical relationships do you know with respect to h, p, b, and z?

Is there a relationship among, c, j, and k? If so. what is it?

Is there a relationship among x, y, and z? If so, what is it?

That makes up nine relationships so you may have enough information to solve the problem. However, it may not be enough (if some of the relationships are redundant). Is there any other relationship given?
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Here is the picture. I finally managed. :)
20180714_183543.jpg
 
OK. The sides are not labelled as I did but let's go with yours. Notice that you did not tell us the areas of the smaller triangles originally.

Notice that line BC is divided by the line with length H. Call the point of intersection D, and call the lengths of BD and CD p and q respectively.

So we have five unknowns and thus need five independent mathematical relationships.

\(\displaystyle h^2 + p^2 = c^2,\ h^2 + q^2 = b^2,\ b^2 + c^2 = a^2,\)

\(\displaystyle p + q = a,\ 0.5 * hp = \sqrt{2}, \ 0.5 * hq = 8 \sqrt{2}, \ 0.5 * bc = 9 \sqrt{2}.\)

Where did I get these?

We have seven obvious relationships, but some are probably dependent. What next?
 
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happybeans said:
Here is the picture. I finally managed. :)
View attachment 9756
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Presumably the problem is to find all four lengths?

One very useful fact for problems with this figure is that the three right triangles you can see are all similar. Can you see why?

Another is that the ratios of areas of similar figures is the square of the ratio of their sides. Among other things, this allows you to find the ratio of b and c.

You might then start with your original question: What equation can you write involving b and c from the fact that the total area is 9 sqrt(2)?

From these two equations, you could find the answer. There are very many other ways to proceed.

Be aware that the figure is not at all to scale -- what looks like the smaller area is actually 8 times as large! So don't expect things to be as they look.
 
I found all the relations so far, but I do not know how to find on side; for example b.
 
happybeans said:
I found all the relations so far, but I do not know how to find on side; for example b.
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Can you show us what you have done, so we can help you along from there?

The method I suggested involves solving a system of equations for b and c.
 
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