SOLUTION: If the shortest side of a right triange is tripled, with 4 added to the result, the result is the longest side. The medium side is 10 more than twice the shortest side. Find the le

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Question 138952: If the shortest side of a right triange is tripled, with 4 added to the result, the result is the longest side. The medium side is 10 more than twice the shortest side. Find the length of the shortest side.
Found 2 solutions by solver91311, ankor@dixie-net.com:
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!
Short side: a
Longest side: c (the hypotenuse)
Medium side: b

Since it is a right triangle:

But , and

So:

Expand the binomials, collect like terms on the left, set the whole thing equal to zero, and then solve the quadratic. Hint: the quadratic factors, but factor out a 4 before you start looking for the binomial factors. Hint: One of your roots will be negative, exclude it because you are looking for a positive measure of length.

Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
If the shortest side of a right triangle is tripled, with 4 added to the result, the result is the longest side. The medium side is 10 more than twice the shortest side. Find the length of the shortest side.
;
Let the original triangle sides: a, b, & c(hypotenuse)
:
"If the shortest side of a right triangle is tripled, and 4 added to the result, it's the longest side"
3a + 4 = c
:
"The medium side is 10 more than twice the shortest side."
b = 2a + 10
:
Find the length of the shortest side.
Using a^2 + b^2 = c^2, substitute for b and c, using the 1st two equations:
:
a^2 + (2a+10)^2 = (3a+4)^2
FOIL
a^2 + 4a^2 + 40a + 100 = 9a^2 + 24a + 16
Combine like terms on the left:
a^2 + 4a^2 - 9a^2 + 40a - 24a + 100 - 16 = 0
:
-4a^2 + 16a + 84 = 0
;
Simplify, divide equation by -4
a^2 - 4a - 21 = 0
Factor this to:
(a-7)(a+3) = 0
:
We want the positive solution:
a = +7 is the shorter side
:
:
Using the 1st two equations we have: b = 24, c = 25
:
Check:
7^2 + 24^2 = 25^2
49 + 576 = 625; confirms our solution