A right triangle has 1 leg that is 2" longer than the other leg. the length of the hypotenuse is sqrt 130" Find lengths of legs
0=2x^+4x-126
short leg --- x
longer leg -- x+2
x^2 + (x+2)^2 = (√136)^2
x^2 + x^2 + 4x + 4 = 136
2x^2 + 4x - 132 = 0
x^2 + 2x - 66 = 0
I will complete the square:
x^2 + 2x + .... = 66 + ...
x^2 + 2x + 1 = 66 + 1
(x+1)^2 = 67
x+1 = ±√67
x = -1 + √67 , since x > 0
so the short leg = √67 - 1 , and the longer is √67+1
Factor to solve for x, which cannot be negative.
126 = 3 * 2 * 7 * 3 = 18 * -7
Work it from there.
In contrast to Reiny, I factor to get
(2x+18)(x-7) = 0
Since the distance cannot be negative:
x = 7
x + 2 = 9
thank you for all the help
Where does the 126 come from?
the original question had √136
I saw 130, which minus 4 = 126.
To find the lengths of the legs of the right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Let's denote one leg as x and the other leg as (x + 2). The hypotenuse can be denoted as √130.
We can set up the equation using the Pythagorean theorem:
a^2 + b^2 = c^2
x^2 + (x + 2)^2 = √130^2
Simplifying the equation:
x^2 + (x^2 + 4x + 4) = 130
Combining like terms:
2x^2 + 4x + 4 = 130
Rearranging the equation:
2x^2 + 4x - 126 = 0
Now we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 4, and c = -126.
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
x = (-4 ± √(4^2 - 4(2)(-126))) / (2(2))
Simplifying:
x = (-4 ± √(16 + 1008)) / 4
x = (-4 ± √1024) / 4
x = (-4 ± 32) / 4
We have two possible solutions:
1) x = (-4 + 32) / 4 = 28 / 4 = 7
2) x = (-4 - 32) / 4 = -36 / 4 = -9
Since the lengths of the sides of a triangle cannot be negative, we discard the negative solution.
Therefore, the length of the leg (x) is 7 inches, and the length of the other leg (x + 2) is 7 + 2 = 9 inches.
Thus, the lengths of the legs of the right triangle are 7 inches and 9 inches.