The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments with lengths in the ratio 1 colon 2. The length of the altitude is 8. How long is the hypotenuse?
(1 point)
Responses
16
16
24
24
4 StartRoot 2 EndRoot
Image with alt text: 4 StartRoot 2 EndRoot
12 StartRoot 2 EndRoot
Let the lengths of the two segments of the hypotenuse be x and 2x.
By the Pythagorean theorem, we have:
x^2 + 8^2 = (2x)^2
x^2 + 64 = 4x^2
3x^2 = 64
x^2 = 64/3
x = 8/√3
The length of the hypotenuse is x + 2x = 3x = 3(8/√3) = 24/√3.
To rationalize the denominator, we multiply the numerator and denominator by √3:
(24/√3)(√3/√3) = (24√3) / 3 = 8√3.
Therefore, the length of the hypotenuse is 8√3 or 24.