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

Question

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

Allison Allen · Accepted Answer

Coordinates of orthocenter #(9/11, -47/11)# #Let# #A = (5,2)# #Let# #B = (3,7)# #Let# #C = (0,9)# Equation for altitude through A: #x(x_3-x_2)+y(y_3-y_2)=x_1(x_3-x_2)+y1(y_3-y_2)# #=>x(0-3)+y(9-7)=(5)(0-3)+(2)(9-7)# #=>-3x + 2y = -15 + 4# #=>color(red)(3x - 2y + 11 = 0)#-----(1) Equation for altitude through B: #x(x_1-x_3)+y(y_1-y_3)=x_2(x_1-x_3)+y2(y_1-y_3)# #=>x(5-0)+y(2-9)=(3)(5-0)+(7)(2-9)# #=>5x -7y=15-49# #=>color(blue)(5x - 7y -34 = 0#-----(2) Equating (1) & (2): #color(red)(3x - 2y +1 1 =color(blue)(5x - 7y -34)# #=>color(orange)(y=-47/11)#-----(3) Plugging (3) in (2): #=>color(violet)(x= 9/11# The orthocenter is at #(9/11, -47/11)# which is actually outside the #triangle# because the #triangle# is an obtuse one#

Sadie Allen · Answer

undefined 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.