SOLUTION: Solve the following word problem. The hypotenuse of a right triangle is four feet longer than three times the shorter leg. The longer leg is one foot less than the hypotenuse.

Question 77304: Solve the following word problem.
The hypotenuse of a right triangle is four feet longer than three times the shorter leg. The longer leg is one foot less than the hypotenuse. Find the dimensions of the right triangle.
Found 2 solutions by jim_thompson5910, checkley75:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let a=shorter leg, b=longer leg, c=hypotenuse
So we have
"The hypotenuse of a right triangle is four feet longer than three times the shorter leg"
"The longer leg is one foot less than the hypotenuse"
Now solve for a:

Plug into a and plug in into b of the Pythagorean theorem

foil the terms on the left side
Combine like terms
Subtract from both sides
Now use the quadratic formula to solve for c (note:this solver uses decimals instead of fractions):

Solved by pluggable solver: SOLVE quadratic equation with variable

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=7.11111111111112 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 25, 1. Here's your graph:

So the hypotenuse is:
c=1 or c=25
Use this to find a
Let c=1

Since this length is negative, we must disregard the hypotenuse length of 1
Let c=25

Now use c=25 to find b

So our dimensions are:
a=7,b=24,c=25

Check:
Plug in a=7, b=24, c=25

works
plug in a=7 and c=25

works

works
Answer by checkley75(3666) (Show Source): You can put this solution on YOUR website!
HYP=(3X+4) X BEING THE SHORTEST LEG1.
LONGER LEG2=(3X+4)-1
THUS THE HYP^2=(LEG1)^2+(LEG*2)^2
(3X+4)^2=X^2+(3X+4-1)^2
9X^2+24X+16=X^2+(3X+3)^2
9X^2+24X+16=X^2+9X^2+18X+9 WE CAN CANCEL OUT THE 9X^2 TERMS & WE HAVE LEFT
24X-X^2-18X+16-9=0
-X^2+6X+7=0
X^2-6X-7=0
(X-7)(X+1)=0
X-7=0
X=7 ANSWER.
HYP=3*7+4
HYP=21+4
HYP=25 ANSWER.
LONG SIDE=(3*7+3)
LONG SIDE=24 ANSWER.
SHORT SIDE=7 ANSWER.
25^2=24^2+7^2
625=576+49
625=625
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