Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the orthocenter of triangle XYZ

Answer 1

First, let's find the equations of the lines that contain the sides of triangle XYZ.

The line containing side XY passes through the points X(2, 5) and Y(8, 5). Since the y-coordinates are the same for both points, the line must be a horizontal line with equation y = 5.

The line containing side YZ passes through the points Y(8, 5) and Z(8, 1). Since the x-coordinates are the same for both points, the line must be a vertical line with equation x = 8.

The line containing side ZX passes through the points Z(8, 1) and X(2, 5). To find the equation of this line, we can use the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line.

The slope m can be found using the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line. Plugging in the coordinates of Z(8, 1) and X(2, 5), we get m = (5 - 1) / (2 - 8) = 4 / (-6) = -2 / 3.

Choosing Z(8, 1) as the point (x₁, y₁), we can plug in the slope and point into the point-slope form: y - 1 = -(2/3)(x - 8).

Simplifying, we get y - 1 = -(2/3)x + 16/3.

Rearranging the equation, we get y = -(2/3)x + 19/3.

Now, let's find the intersection points of the lines.

The horizontal line y = 5 intersects the vertical line x = 8 at the point (8, 5).

The horizontal line y = 5 intersects the line y = -(2/3)x + 19/3 at the point where y = 5. Plugging in y = 5 into the equation, we get 5 = -(2/3)x + 19/3. Multiplying through by 3 to clear the fraction, we get 15 = -2x + 19. Solving for x, we get x = -4. Plugging this into the line equation, we get y = -(2/3)(-4) + 19/3 = 23/3.

Therefore, the intersection point of the horizontal line y = 5 and the line y = -(2/3)x + 19/3 is (-4, 23/3).

The vertical line x = 8 intersects the line y = -(2/3)x + 19/3 at the point where x = 8. Plugging in x = 8 into the equation, we get y = -(2/3)(8) + 19/3 = 17/3.

Therefore, the intersection point of the vertical line x = 8 and the line y = -(2/3)x + 19/3 is (8, 17/3).

The coordinates of the orthocenter of triangle XYZ are the intersection points of the altitudes of the triangle. Since the altitudes of a triangle are perpendicular to the sides, the intersection points are the vertices of a perpendicular triangle. In other words, the orthocenter is the intersection point of line XY and line YZ, which is (-4, 23/3).