# What is the orthocenter of a triangle with corners at #(5 ,2 )#, #(3 ,7 )#, and (0 ,9 )#?

Coordinates of orthocenter

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To find the orthocenter of a triangle, we need to find the point where the altitudes of the triangle intersect. An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side.

Step 1: Find the slopes of the sides of the triangle. Step 2: Find the equations of the perpendicular bisectors of the sides. Step 3: Find the intersection point of any two perpendicular bisectors. Step 4: This intersection point is the orthocenter of the triangle.

Given the coordinates of the triangle's vertices: A (5, 2) B (3, 7) C (0, 9)

Step 1: Slopes of the sides of the triangle: Slope of AB: ( m_{AB} = \frac{7 - 2}{3 - 5} = -\frac{5}{2} ) Slope of AC: ( m_{AC} = \frac{9 - 2}{0 - 5} = -\frac{7}{5} ) Slope of BC: ( m_{BC} = \frac{9 - 7}{0 - 3} = -\frac{2}{3} )

Step 2: Equations of perpendicular bisectors: Equation of the perpendicular bisector of AB: Midpoint of AB: ( (\frac{5 + 3}{2}, \frac{2 + 7}{2}) = (4, 4.5) ) Slope of AB perp: ( m_{ABperp} = \frac{1}{m_{AB}} = -\frac{2}{5} ) Equation: ( y - 4.5 = -\frac{2}{5}(x - 4) )

Equation of the perpendicular bisector of AC: Midpoint of AC: ( (\frac{5 + 0}{2}, \frac{2 + 9}{2}) = (2.5, 5.5) ) Slope of AC perp: ( m_{ACperp} = \frac{1}{m_{AC}} = -\frac{5}{7} ) Equation: ( y - 5.5 = -\frac{5}{7}(x - 2.5) )

Equation of the perpendicular bisector of BC: Midpoint of BC: ( (\frac{3 + 0}{2}, \frac{7 + 9}{2}) = (1.5, 8) ) Slope of BC perp: ( m_{BCperp} = \frac{1}{m_{BC}} = -\frac{3}{2} ) Equation: ( y - 8 = -\frac{3}{2}(x - 1.5) )

Step 3: Find the intersection point of any two perpendicular bisectors. For example, we can find the intersection of the perpendicular bisectors of AB and AC.

Solving ( y = -\frac{2}{5}(x - 4) + 4.5 ) and ( y = -\frac{5}{7}(x - 2.5) + 5.5 ), we find the intersection point as (2, 3).

Step 4: The intersection point (2, 3) is the orthocenter of the triangle.

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