Question

Find the coordinates of the orthocenter of angle XYZ with vertices X(-3, 3), Y(9, 3), and Z(1, -5).

The coordinates of the orthocenter are(__,__)

Answer 1

The orthocenter of a triangle is the intersection point of the three altitudes of the triangle.

An altitude of a triangle is a line that goes from one vertex of the triangle to the opposite side, and is perpendicular to that side.

To find the orthocenter of a triangle, we can find the equation of each altitude by using the slope-point form.

The slope of the altitude from vertex X to side YZ is the negative reciprocal of the slope of line YZ.

The slope of line YZ is (3 - (-5)) / (9 - 1) = 8/8 =1

So the slope of the altitude from vertex X is -1

The equation of this altitude is y = -1x + b

We know that this line passes through the point X (-3, 3) so we can substitute this point in the equation and find the value of b

3 = -1(-3) + b

b = 6

So the equation of this altitude is y = -1x + 6

The orthocenter of angle XYZ is the point that is common to all three altitudes, we can find this point by finding the intersection point of all three altitudes.

So the coordinates of the orthocenter of angle XYZ are (1, -1)