Question

A right triangle is drawn on a coordinate plane with a line from the right angle that is perpendicular to the hypotenuse. The hypotenuse consists of a short line segment measuring 2 units and a long line segment measuring 30 units.

Answer 1

Do you want a specific quantity (leg lengths, altitude, area, coordinates)? Common useful results:

Let the hypotenuse segments be m = 2 and n = 30, so the hypotenuse c = m + n = 32. For a right triangle with altitude from the right angle to the hypotenuse:

- leg adjacent to m: a = sqrt(c·m) = sqrt(32·2) = sqrt64 = 8
- leg adjacent to n: b = sqrt(c·n) = sqrt(32·30) = sqrt960 = 8√15
- altitude from the right angle: h = sqrt(mn) = sqrt(60) = 2√15
- area: (1/2)ab = (1/2)·8·8√15 = 32√15 (equivalently (1/2)·c·h = (1/2)·32·2√15)

A convenient placement on coordinates: take hypotenuse endpoints at A = (0,0) and B = (32,0) and the foot of the altitude at F = (2,0). Then the right-angle vertex is R = (2, 2√15), giving legs AR = 8 and BR = 8√15, as above.

If you want a step-by-step derivation or another quantity (perimeter, angles, etc.), tell me which.