Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the orthocenter of triangle XYZ.(1 point) Responses (6,4) (6,4) (8,5) (8,5) (5,6) (5,6) (5,3)
To find the orthocenter of a triangle, you need to find the intersection point of its altitudes.
To find the altitude, you take two sides of the triangle and find the slope of the line perpendicular to that side. Then, you find the equation of the line passing through the third vertex with that slope.
Let's find the equation of the altitude for each side of triangle XYZ:
Side XY:
Slope of XY = (5 - 5) / (2 - 8) = 0
The line perpendicular to XY will have an undefined slope, so it will be vertical and pass through the midpoint of XY, which is ((2+8)/2, (5+5)/2) = (5, 5).
Side XZ:
Slope of XZ = (1 - 5) / (8 - 2) = -0.5
The line perpendicular to XZ will have a slope of 2 (negative reciprocal of -0.5) and pass through the midpoint of XZ, which is ((2+8)/2, (5+1)/2) = (5, 3).
Side YZ:
Slope of YZ = (1 - 5) / (8 - 8) = undefined
The line perpendicular to YZ will have a slope of 0, so it will be horizontal and pass through the midpoint of YZ, which is ((8+8)/2, (5+1)/2) = (8, 3).
Now, we need to find the point of intersection of the three altitudes. Since we have a vertical and horizontal line, we can directly find that the point of intersection is (5, 3).
Therefore, the coordinates of the orthocenter of triangle XYZ are (5, 3).
Answer: (5, 3)