what are the coordinates of the orthocenter of the triangle with vertices at A(4,-2), B(4,6), and C(-2,4)

Answer 1

Orthocenter = intersection of the altitudes of the triangle, which are lines from each vertex, perpendicular to the opposite side.

Start by plotting the 3 points on graph paper. Call the angles A, B, and C (same as the point designations), and the sides a, b, and c, where each is opposite the angle with the same letter---i.e. "a" is the side connecting B and C---etc.

We can immediatley draw on altitude: from C to point (4,4) on line c, with length = 6. This in turn gives us side a, using the Pythagorean Theorem: a^2 = 6^2 + 2^2 = 40, a = 2√10

We can also get side b using the Pythagorean Theorem: b^2 = 6^2 + 6^2, b = 6√2

Next, get the angle B using tan(B) = 6/2.

The rest of the lengths and angles are easily found with the usual trig formulas.