SOLUTION: a right triangle has hypotenuse of lenght 13 and legs A and B. If the area of the triangle is 14, what is the sum A+B? We do not think this is possible. If the Square of the hyp

Algebra ->  Pythagorean-theorem -> SOLUTION: a right triangle has hypotenuse of lenght 13 and legs A and B. If the area of the triangle is 14, what is the sum A+B? We do not think this is possible. If the Square of the hyp      Log On


   

Question 261102: a right triangle has hypotenuse of lenght 13 and legs A and B. If the area of the triangle is 14, what is the sum A+B? We do not think this is possible. If the Square of the hypotenuse is 169, and it is an equilateral triangle, the sides would be about 9, and one half of 9*9 (the area of the triangle) is more like 40 than 14
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
a right triangle has hypotenuse of lenght 13 and legs A and B. If the area of the triangle is 14, what is the sum A+B?
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a^2 + b^2 = 169
ab/2 = 14 --> a = 28/b
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(28/b)^2 + b^2 = 169
784 + b^4 = 169b^2
b^4 - 169b^2 + 784 = 0
Sub x for b^2
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-169x%2B784+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-169%29%5E2-4%2A1%2A784=25425.

Discriminant d=25425 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--169%2B-sqrt%28+25425+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-169%29%2Bsqrt%28+25425+%29%29%2F2%5C1+=+164.22609359551
x%5B2%5D+=+%28-%28-169%29-sqrt%28+25425+%29%29%2F2%5C1+=+4.77390640449013

Quadratic expression 1x%5E2%2B-169x%2B784 can be factored:
1x%5E2%2B-169x%2B784+=+%28x-164.22609359551%29%2A%28x-4.77390640449013%29
Again, the answer is: 164.22609359551, 4.77390640449013. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-169%2Ax%2B784+%29

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b^2 =~ 164.226
b^2 =~ 4.7739
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b =~ 12.8166, a =~ 2.1849 or vice versa
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a + b = 15
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In any right triange with a specified lenght of the hypotenuse, the max area is
c^2/2.
For a hyp = 13, that's 169/2 = 84.5 sq units. Any area less than that can be made with hyp = 13.