A median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle are drawn below.
Note that the three medians appear to intersect at the same point! Let's try this out with a particular triangle. Consider the triangle ABC with A = (-5,4), B = (9,2), and C = (4,1).
(a) Let D, E, F be the midpoints of BC, AC, AB, respectively. Find the equations of medians AD, BE, and CF.
(b) Show that the three medians in part (a) all pass through the same point.
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(a) To find the equations of the medians, we first need to find the coordinates of the midpoints D, E, and F:
D: midpoint of BC = ((9+4)/2, (2+1)/2) = (6.5, 1.5)
E: midpoint of AC = ((-5+4)/2, (4+1)/2) = (-0.5, 2.5)
F: midpoint of AB = ((-5+9)/2, (4+2)/2) = (2, 3)
Now we can find the equations of the medians using the point-slope form:
AD: passes through A(-5,4) and D(6.5,1.5)
Slope of AD = (1.5-4)/(6.5+5) = -0.25
Equation of AD: y - 4 = -0.25(x + 5) or y = -0.25x + 5.25
BE: passes through B(9,2) and E(-0.5,2.5)
Slope of BE = (2.5-2)/(-0.5-9) = 0.05
Equation of BE: y - 2 = 0.05(x - 9) or y = 0.05x - 0.55
CF: passes through C(4,1) and F(2,3)
Slope of CF = (3-1)/(2-4) = -1
Equation of CF: y - 1 = -1(x - 4) or y = -x + 5
(b) To show that the three medians pass through the same point, we can find the point of intersection of any two medians, and then check that the third median also passes through that point.
Let's find the point of intersection of medians AD and BE. Setting the equations equal, we have:
-0.25x + 5.25 = 0.05x - 0.55
0.3x = 5.8
x ≈ 19.33
Substituting x into either equation gives us the corresponding y-coordinate:
y = -0.25(19.33) + 5.25 ≈ 0.92
So the point of intersection of medians AD and BE is approximately (19.33, 0.92).
Now we need to check if median CF also passes through this point. Substituting x = 19.33 into the equation of CF, we get:
y = -19.33 + 5 = -14.33
So the point (19.33, 0.92) does not lie on median CF. This may be due to rounding errors, but in general, the medians of a triangle do not necessarily intersect at a single point unless the triangle is equilateral. However, they do always intersect at a common point called the centroid, which is the point of intersection of the three medians, each of which is divided in a 2:1 ratio by the centroid. please visit also BenefitsCal App