Answer:
The side lengths are 5, 12, 13
Step-by-step explanation:
These are the side lengths we know before we have solved the problem.
a= x b= x+7 c= 13
If we put this into the Pythagorean theorem, we get
[tex]13^{2} = x^{2} +(x+7)^{2}[/tex]
Expand
[tex]169 = 2x^2+14x+49\\[/tex]
Subtract 169 from both sides
[tex]0 =2x^2+14x-120[/tex]
Solve using the quadratic formula
a= 2 b= 14 c= -120
[tex]\frac{-14\pm \sqrt{14^2-4\cdot \:2\left(-120\right)}}{2\cdot \:2}[/tex]
[tex]x= \frac{-14\pm \:34}{2\cdot \:2}[/tex]
If we separate the solutions we get
[tex]x= \frac{20}{4} = 5[/tex]
and
[tex]x= \frac{-48}{4} = -12[/tex]
Since this is a question about length, we can ignore the negative.
So the longer leg is 12ft and the shorter leg is 5ft
The shorter length is 5feet and the longer length is 12 feet
Let the shorter leg = x
Let the longer leg = x+7
Therefore,
x² + (x+7)² = 13²
x² + x² + 14x + 49 = 169
Collect like terms
2x² + 14x + 49 - 169 = 0
2x² + 14x - 120 = 0.
Divide by 2
x² + 7x - 60
By using quadratic equation
x² + 12x - 5x - 60
x(x + 12) - 5(x + 12)
Therefore, x - 5 = 0
x = 0 + 5.
x = 5
The shorter length is 5ft
Longer length = 5+7 = 12 feet
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