Question 148002
Hi, Hope I can help
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The longer leg of a right triangle is 1 cm. longer than the shorter leg and the hypontenuse is 5cm, then what are the lengths of the leg?
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The shorter leg is represented by "x", the longer leg is (x+1)
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We will use the Pythagorean theorem to solve for the two sides, the Pythagorean theorem equation is
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{{{ a^2 + b^2 = c^2 }}}
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"a" and "b" are the sides, "c" is the hypontenuse. We will now replace "a", "b" , and "c"
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{{{ (x+1)^2 + x^2 = 5^2 }}}
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{{{ (x+1)^2 + x^2 = 25 }}}
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We will solve it even more
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{{{ (x^2 + 2x +1) + x^2 = 25 }}}
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{{{ x^2 + 2x +1 + x^2 = 25 }}}
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{{{ 2(x)^2 + 2x +1 = 25 }}}
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We will move the "25" to the left side
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{{{ 2(x)^2 + 2x +1 -25 = 25 - 25 }}}
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{{{ 2(x)^2 + 2x - 24 = 0 }}}
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We can reduce the equation, we will divide everything by "2"
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{{{ (2(x)^2 + 2x - 24)/2 = 0/2 }}}
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{{{ x^2 + x - 12 = 0 }}}
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We can use the quadratic equation
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{{{ x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
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This is the original equation using "a", "b", and "c"
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{{{ a(x)^2 + (b)x + c = 0 }}}
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{{{ x^2 + x - 12 = 0 }}}
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a = 1, b = 1, c = -12
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Now we can replace them in the quadratic equation,
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{{{ x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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{{{ x = (-1 +- sqrt( 1^2-4*1*(-12) ))/(2*1) }}} 
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{{{ x = (-1 +- sqrt( 1-(-48) ))/2 }}} 
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{{{ x = (-1 +- sqrt( 1 + 48))/2 }}} 
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{{{ x = (-1 +- sqrt (49) )/2 }}}
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{{{ x = (-1 +- 7 )/2 }}}
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We will use addition(if we used subtraction, the answer would be negative)
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{{{ x = (-1 + 7 )/2 }}}
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{{{x = 6/2 }}}
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{{{x = 3 }}}
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x = 3 cm.( lengths or measurements can't be negative, the other answer would be (-4) )( you can check by replacing "x" in {{{ (x+1)^2 + x^2 = 25 }}} with "3"
or "-4"
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The shorter side was x, or 3 cm., the longer side was (x+1), or 4 cm.
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You can check by replacing "a" and "b" in the Pythagorean theorem equation
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{{{ a^2 + b^2 = c^2 }}}
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{{{ 4^2 + 3^2 = 5^2 }}}
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{{{ 16 + 9 = 25 }}}
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{{{ 25 = 25 }}}
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Shorter side = 3 cm.
Longer side = 4 cm.
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Hope I helped, Levi