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The Orthocentre of a Triangle
The orthocentre of a triangle is the point of intersection of the three altitudes of the triangle. An altitude is a line segment which goes from a vertex to the opposite side of the triangle meeting at 900 (ie: perpendicular). Sometimes the orthocentre is inside the triangle (acute triangles) and sometimes it is outside the triangle (obtuse triangles). http://www.mathopenref.com/triangleorthocenter.html
To find the coordinates of the orthocentre of a triangle we must find the equations of the altitudes of each side of the triangle.
• Find the slopes of each side of the triangle
• Determine the perpendicular slopes to each side of the triangle (ie: negative reciprocals)
• Using the vertex and the perpendicular slopes, determine the equation of the altitudes for each side of the triangle
• Using substitution or elimination with two of the median equations, determine the point of intersection
• Check this point in the third equation
Example # 1:
Triangle ABC has vertices A (0,0) , B (6,4) and C (12,-4). Determine
the equation of BK, the altitude from B to AC.
y
8
7
6
5
B
4
3
2
1
x
A
­2 ­1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
­1
­2
K
­3
­4
­5
­6
C
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Example # 2:
Find the coordinates of the orthocentre using the triangle from
example # 1.
Equation of Altitude from B to AC we found in example # 1:
y
8
7
6
5
4
B
3
2
1
x
A
­2 ­1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
­1
­2
­3
­4
C
­5
­6
Check in 3x ­ 4y = 0
Therefore the orthocentre of triangle ABC is Example # 3: Verify that the coordinates of the orthocentre for triangle ABC with
vertices A (-1,2) , B (3,5) and C (7,-2) is
y
10
9
8
7
6
5
4
3
2
1
x
­10­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
­2
­3
­4
­5
­6
­7
­8
­9
­10
Homework: p. 194 # 4 ­ 7 , 14