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.

Algebra ->  Algebra  -> Pythagorean-theorem -> 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.       Log On

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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(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Let a=shorter leg, b=longer leg, c=hypotenuse
So we have
c=3a%2B4 "The hypotenuse of a right triangle is four feet longer than three times the shorter leg"
b=c-1 "The longer leg is one foot less than the hypotenuse"
Now solve for a:
c=3a%2B4
c-4=3a
a=%281%2F3%29c-4%2F3
Plug %281%2F3%29c-4%2F3 into a and plug in c-1 into b of the Pythagorean theorem
%28%281%2F3%29c-4%2F3%29%5E2%2B%28c-1%29%5E2=c%5E2
%28%281%2F9%29c%5E2-%288%2F9%29c%2B16%2F9%29%2B%28c%5E2-2c%2B1%29=c%5E2 foil the terms on the left side
%28%2810%2F9%29c%5E2-%2826%2F9%29c%2B25%2F9%29=c%5E2 Combine like terms
%28%281%2F9%29c%5E2-%2826%2F9%29c%2B25%2F9%29=0 Subtract c%5E2 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 ac%5E2%2Bbc%2Bc=0 (in our case 0.111111111111111c%5E2%2B-2.88888888888889c%2B2.77777777777778+=+0) has the following solutons:

c%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=%28-2.88888888888889%29%5E2-4%2A0.111111111111111%2A2.77777777777778=7.11111111111112.

Discriminant d=7.11111111111112 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2.88888888888889%2B-sqrt%28+7.11111111111112+%29%29%2F2%5Ca.

c%5B1%5D+=+%28-%28-2.88888888888889%29%2Bsqrt%28+7.11111111111112+%29%29%2F2%5C0.111111111111111+=+25
c%5B2%5D+=+%28-%28-2.88888888888889%29-sqrt%28+7.11111111111112+%29%29%2F2%5C0.111111111111111+=+1

Quadratic expression 0.111111111111111c%5E2%2B-2.88888888888889c%2B2.77777777777778 can be factored:
0.111111111111111c%5E2%2B-2.88888888888889c%2B2.77777777777778+=+0.111111111111111%28c-25%29%2A%28c-1%29
Again, the answer is: 25, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+0.111111111111111%2Ax%5E2%2B-2.88888888888889%2Ax%2B2.77777777777778+%29


So the hypotenuse is:
c=1 or c=25
Use this to find a
Let c=1
1=3a%2B4
a=-1
Since this length is negative, we must disregard the hypotenuse length of 1
Let c=25
25=3a%2B4
a=7
Now use c=25 to find b
b=25-1
b=24
So our dimensions are:
a=7,b=24,c=25


Check:
a%5E2%2Bb%5E2=c%5E2 Plug in a=7, b=24, c=25
7%5E2%2B24%5E2=25%5E2
49%2B576=625
625=625 works
c=3a%2B4 plug in a=7 and c=25
25=3%287%29%2B4
25=25 works
b=c-1
24=25-1
24=24 works


Answer by checkley75(3666) About Me  (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