Consider ΔABC shown below.
The following steps would be useful to find the equation of the altitude AD.
Step 1 :
Find the slope of BC.
Step 2 :
Since AB and BC are perpendicular,
slope of AD x slope of BC = -1
slope of AD = -1/slope of BC
Step 3 :
Altitude AD is passing through the point A. In slope intercept form equation of a line 'y = mx + b', using the slope of AD and point A, find the y-intercept 'b'.
Step 4 :
Write the equation of the altitude AD using the slope of AD, 'm' and y-intercept 'b'.
Similarly, we can find the equations of altitudes through the vertices B and C.
Example :
A(-3, 0) B(10, -2) and C(12, 3) are the vertices of ΔABC. Find the equations of the altitudes through A, B and C in general form.
Solution :
Equation of the altitude through A :
slope of BC = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (x_{1}, y_{1}) = B(10, -2) and (x_{2}, y_{2}) = C(12, 3).
= (3 + 2)/(12 - 10)
slope of BC = 5/2
slope of AD = -1/slope of BC
= -1/(5/2)
= -2/5
Equation of the altitude AD :
y = mx + b
Substitute m = 5/2.
y = (-2/5)x + b ----(1)
Substitute (x, y) = A(-3, 0).
0 = (-2/5)(-3) + b
0 = 6/5 + b
-6/5 = b
(1)----> y = (-2/5)x - 6/5
Multiply each side by 5.
5y = -2x - 6
2x + 5y + 6 = 0
Equation of the altitude through B :
slope of AC = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (x_{1}, y_{1}) = A(-3, .0) and (x_{2}, y_{2}) = C(12, 3).
= (3 - 0)/(12 + 3)
= 3/15
slope of AC = 1/5
slope of BE = -1/slope of AC
= -1/(1/5)
= -5
Equation of the altitude BE :
y = mx + b
Substitute m = -5.
y = -5x + b ----(2)
Substitute (x, y) = B(10, -2).
-2 = -5(10) + b
-2 = -50 + b
48 = b
(2)----> y = -5x + 48
5x + y - 48 = 0
Equation of the altitude through C :
slope of AB = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute (x_{1}, y_{1}) = A(-3, 0) and (x_{2}, y_{2}) = B(10, -2).
= (-2 - 0)/(10 + 3)
= -2/13
slope of AB = -2/13
slope of CF = -1/slope of AB
= -1/(-2/13)
= 13/2
Equation of the altitude CF :
y = mx + b
Substitute m = 13/2.
y = (13/2)x + b ----(3)
Substitute (x, y) = C(12, 3).
3 = (13/2)(12) + b
3 = 78 + b
-75 = b
(3)----> y = (13/2)x - 75
Multiply each side by 2.
2y = 13x - 150
-13x + 2y + 150 = 0
13x - 2y - 150 = 0
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