SOLUTION: The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into two segments, whose lengths are 8" and 18". How long is the altitude?

Algebra ->  Formulas -> SOLUTION: The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into two segments, whose lengths are 8" and 18". How long is the altitude?       Log On


   

Question 149702: The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into two segments, whose lengths are 8" and 18". How long is the altitude?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's draw the picture. Be sure you label every length you can


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So we can see that the hypotenuse of the largest triangle is 26 units. Using pythagoreans theorem, we get:

x%5E2%2By%5E2=26%5E2


Now looking at the left most smaller triangle, we see that the legs are 8 and "h" while the hypotenuse is "x". So once again, with pythagoreans theorem, we can say:

8%5E2%2Bh%5E2=x%5E2


Now solving for h%5E2, we get h%5E2=x%5E2-8%5E2


Finally, the right most smaller triangle has a base of 18 and "h" and a hypotenuse of y. So this gives us:

h%5E2%2B18%5E2=y%5E2

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x%5E2%2By%5E2=26%5E2 Start with the first equation.


x%5E2%2B%28h%5E2%2B18%5E2%29=26%5E2 Plug in h%5E2%2B18%5E2=y%5E2. In other words, replace y%5E2 with h%5E2%2B18%5E2


x%5E2%2Bh%5E2%2B324=676 Square 18 to get 324. Square 26 to get 676.


x%5E2%2Bh%5E2%2B324=676 Square 18 to get 324. Square 26 to get 676.


x%5E2%2Bx%5E2-8%5E2%2B324=676 Plug in h%5E2=x%5E2-8%5E2


x%5E2%2Bx%5E2-64%2B324=676 Square 8 to get 64


2x%5E2%2B260=676 Combine like terms.


2x%5E2=416 Subtract 260 from both sides.


x%5E2=208 Divide both sides by 2.


x=sqrt%28208%29 Take the square root of both sides. Note: only the positive square root is considered.


So the length of x is x=sqrt%28208%29 which is about 14.422 units (this isn't important).


h%5E2=x%5E2-8%5E2 Go back to the second equation


h%5E2=%28sqrt%28208%29%29%5E2-8%5E2 Plug in x=sqrt%28208%29


h%5E2=208-8%5E2 Square the square root to eliminate it.


h%5E2=208-64 Square 8 to get 64


h%5E2=144 Combine like terms.


h=12 Take the square root of both sides. Once again, only the positive square root is considered.

So the height is 12 units which means that altitude is 12 units.