+1234 A median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. Let's try this out with a particular triangle. Consider the triangle ABC with A=(0, 0) B=(1, 0) C=(0, 1).
(a) Let D, E, F be the midpoints of BC, AB, AC respectively. Find the equations of medians AD, BE, CF and
(b) Show that the three medians in part (a) all pass through the same point.
+1234 Midpoint BC = [ (-9 + 1)/2 , (6 - 4) / 2 ] = [ -4, 1]
Midpoint AB = [ (5 -9)/2 , (4 + 6) / 2 ] = [ -2, 5]
Midpoint AC = [ (5 + 1) / 2 , (4 -4) / 2 ] = [ 3 , 0 ]
Slope AD = [ 4 - 1 ] / [ 5 - -4] = 1/3
Slope CE = [ -4-5] / [ 1 - -2] = -3
Slope BF = [ 6-0] / [ -9-3] = -1/2
Equation of line through AD
y = (1/3)(x + 4) + 1 (1)
Equation of line through CE
y = -3 ( x + 2) + 5 (2)
Equation of line through BF
y = (-1/2)(x -3) (3)
Set (1), (2) equal to find their x intersection
(1/3)(x + 4) + 1 = -3(x + 2) + 5
x + 4 + 3 = -9(x + 2) + 15
x+ 7 = -9x -18 + 15
x + 7 = -9x -3
10x = -10
x = -1
y = -3(-1+2) + 5 = 2
Intersection pt = ( -1 , 2)
Test in (3)
2 = (-1/2) (-1 -3)
2 = 2 true
