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