Posted by Anonymous on Saturday, March 5, 2011 at 2:19pm.
A hypotenuse has a ratio of 1:2 with what? the altitude? itself? its area?
Do you mean the ratio of the altitude to the hypotenuse is 1:2? If so, from (x + y) = 2(6) = 12 and xy = 36, we derive y^2 - 12y + 36 = 0 from which y = 6 and x = 6, an isosceles right triangle with sides of 8.4852, hypotenuse of 12, altitude of 6 and area of 36.
Do you mean the segments of the hypotenuse, defined by where the altitude intersects the hypotenuse, are in the ratio of 1:2? If so, with x and y being the segments of the hypotenuse, x/y = 1/2 and xy = 6^2 = 36 from which (y/2)y = 36 making y^2 = 72 or y = 8.485 and x = 4.242 yielding a triangle with sides of 7.148 and 10.392, hypotensue of 12.727, altitude of 6 and area of 38.181.