Find the lengths of each side give the following right triangle:
Base=x
hypotenuse=x+6
last side=x+3
thank you!
by Pythagoras ... x^2 + (x + 3)^2 = (x + 6)^2
2x^2 + 6x + 9 = x^2 + 12x + 36
x^2 - 6x - 27 = 0 ... (x + 3)(x - 9) = 0
To find the lengths of each side of the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's label the sides of the triangle:
- Base: x
- Hypotenuse: x + 6
- Last side: x + 3
According to the Pythagorean theorem:
(x + 6)^2 = x^2 + (x + 3)^2
Expanding the equation:
x^2 + 12x + 36 = x^2 + x^2 + 6x + 6x + 9
Combining like terms:
x^2 + 12x + 36 = 2x^2 + 12x + 9
Now, we can bring all terms to one side of the equation:
2x^2 + 12x + 9 - x^2 - 12x - 36 = 0
Simplifying further:
x^2 - 27 = 0
Solving for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 0, and c = -27. Plugging in these values:
x = (0 ± √(0^2 - 4(1)(-27))) / (2(1))
Simplifying:
x = ± √108 / 2
x = ± 6√3 / 2
x = ± 3√3
Since a negative length doesn't make sense in this context, we can take the positive value:
x = 3√3
Thus, the lengths of each side of the right triangle are:
Base: x = 3√3
Hypotenuse: x + 6 = 3√3 + 6
Last side: x + 3 = 3√3 + 3
So, the lengths of the sides are:
Base = 3√3
Hypotenuse = 3√3 + 6
Last side = 3√3 + 3