Find the orthocenter of triangle ABC.

A (3,5) B (9,5) C (5,3)

I think it is (9, 5).

What makes you say it is (9,5) ???

To find the orthocenter of a triangle, we need to find the intersection point of the altitudes. The altitudes are the perpendicular lines drawn from the vertices to their opposing sides.

Step 1: Find the slopes of the sides AB, BC, and CA.
The slope of a line can be determined using the formula:
m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

The slope of AB = (5 - 5) / (3 - 9) = 0 / -6 = 0.
The slope of BC = (3 - 5) / (5 - 9) = -2 / -4 = 1/2.
The slope of CA = (3 - 5) / (5 - 3) = -2 / 2 = -1.

Step 2: Determine the slopes of the altitudes.
The slopes of the altitudes are the negative reciprocal of the slopes of the sides. That is, if the slope of a side is m, the slope of the corresponding altitude will be -1/m.

The slope of the altitude corresponding to side AB = -1/0 (undefined).
The slope of the altitude corresponding to side BC = -1/(1/2) = -2.
The slope of the altitude corresponding to side CA = -1/-1 = 1.

Step 3: Find the equations of the altitudes.
Using the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept, we can find the equations of the altitudes. To find the y-intercept, we need a point that lies on the line. We can use the midpoints of the opposite sides as they are always on the altitude.

The midpoint of side AB = ((3 + 9) / 2, (5 + 5) / 2) = (6, 5).
The midpoint of side BC = ((9 + 5) / 2, (5 + 3) / 2) = (7, 4).
The midpoint of side CA = ((5 + 3) / 2, (3 + 5) / 2) = (4, 4).

The equation of the altitude corresponding to side AB passing through the midpoint (6, 5) is y = 5.
The equation of the altitude corresponding to side BC passing through the midpoint (7, 4) is y = -2x + 18.
The equation of the altitude corresponding to side CA passing through the midpoint (4, 4) is y = x - 4.

Step 4: Find the intersection point of the altitudes.
To find the orthocenter, we need to find the common point where all the altitudes intersect. Since the altitude corresponding to side AB has an undefined slope and is a horizontal line, it will intersect the other two altitudes at their y-intercepts.

Substituting the equation of the altitude of side BC into the equation of altitude of side AB, we get:
-2x + 18 = 5
-2x = -13
x = 13/2

Substituting the value of x into the equation of the altitude of side CA, we get:
y = 13/2 - 4
y = 13/2 - 8/2
y = 5/2

Therefore, the orthocenter of triangle ABC is at the point (13/2, 5/2) or (6.5, 2.5).

Hence, your initial assumption that the orthocenter is at (9, 5) is incorrect. The correct coordinates of the orthocenter are (13/2, 5/2) or (6.5, 2.5).