Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the orthocenter
(6,4)
(8,5)
(5,6)
(5,3)
wha the answer
The orthocenter of a triangle is the point of intersection of its altitudes.
To find the altitude of a triangle, we need to find the perpendicular line that passes through one of its vertices and is perpendicular to the opposite side.
Let's find the altitude from vertex X, which is perpendicular to side YZ.
The slope of the line passing through points Y(8, 5) and Z(8, 1) is (1 - 5) / (8 - 8) = -4 / 0.
Since the slope is undefined, it means that the line is vertical and parallel to the y-axis.
Therefore, the altitude from vertex X is the vertical line passing through X(2, 5), which is the line x = 2.
Similarly, we can find the altitudes from vertices Y and Z.
The altitude from vertex Y is the horizontal line passing through Y(8, 5), which is the line y = 5.
The altitude from vertex Z is the line perpendicular to side XY.
The slope of the line passing through points X(2, 5) and Y(8, 5) is (5 - 5) / (2 - 8) = 0 / -6 = 0.
Since the slope is 0, it means that the line is horizontal and parallel to the x-axis.
Therefore, the altitude from vertex Z is the line y = 1.
Now, we need to find the point of intersection of these altitudes, which is the orthocenter.
The altitude passing through X is the line x = 2.
The altitude passing through Y is the line y = 5.
The altitude passing through Z is the line y = 1.
To find the point of intersection, we need to find the values of x and y that satisfy all three equations.
From the first two equations, we can conclude that the x-coordinate of the orthocenter is 2.
From the second and third equations, we can conclude that the y-coordinate of the orthocenter is 1.
Therefore, the coordinates of the orthocenter are (2, 1).
None of the provided answer choices match the calculated coordinates.