SOLUTION: The hypotenuse of a right triangle is 12 inches and the area is 24 square inches. Find the dimension of the triangle, correct to one decimal place.

Question 1040514: The hypotenuse of a right triangle is 12 inches and the area is 24 square inches. Find the dimension of the triangle, correct to one decimal place.
Found 2 solutions by Alan3354, josmiceli:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
The hypotenuse of a right triangle is 12 inches and the area is 24 square inches. Find the dimension of the triangle, correct to one decimal place.
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Area = b*h/2 = 24
Using 12 for the base, h = 4.
The altitude from the right angle = 4.
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Label the 2 parts of the base c & d, and the 2 sides e & f
c + d = 12
e^2 - c^2 = 16
f^2 - d^2 = 16
e^2 + f^2 = 12^2
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Sub for c: c = 12 - d
e^2 - d^2 + 24d - 144 = 16
e^2 = d^2 - 24d + 160
---
Sub for f: f^2 = d^2 + 16
e^2 + d^2 + 16 = 144
--> e^2 = 128 - d^2
----
128 - d^2 = d^2 - 24d + 160
2d^2 - 24d + 32 = 0
d^2 - 12d + 16 = 0

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=80 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 10.4721359549996, 1.52786404500042. Here's your graph:

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The 2 solutions are c & d. Their sum is 12, the hypotenuse.
6 + sqrt(20) and 6 - sqrt(20)
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e^2 = c^2 + 16
= 36 + 20 + 12sqrt(20)

Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
Let = the area of the right triangle in square inches
Let = the base of the right triangle in inches
Let = the height of the right triangle in inches
----------------------------------------------

-------------------
Use pythagorean theorem

Let

Use quadratic formula

and, also:

Since

--------------------
check:

close enough
----------------------

close enough
-----------------------
The triangle is:
12 x 4.3 x 11.2
Hope I got it!
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