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# Analytic geometry

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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.

Mar 5, 2023

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(a) To find the equations of medians AD, BE, and CF, we first need to find the midpoints of the sides of triangle ABC.

The midpoint of BC is:

D = ((9+4)/2, (2+1)/2) = (6.5, 1.5)

The midpoint of AC is:

E = ((-5+4)/2, (4+1)/2) = (-0.5, 2.5)

The midpoint of AB is:

F = ((-5+9)/2, (4+2)/2) = (2, 3)

Now we can find the equations of the medians.

The equation of median AD can be found by finding the midpoint of BC, D, and the vertex A, and then finding the equation of the line passing through those two points.

The slope of the line passing through A and D is:

m = (1.5 - 4) / (6.5 - (-5)) = -0.25

M = ((-5+6.5)/2, (4+1.5)/2) = (-0.75, 2.75)

Therefore, the equation of median AD is:

y - 2.75 = -0.25(x + 0.75)

Simplifying, we get:

y = -0.25x + 3

The equation of median BE can be found in a similar way. The slope of the line passing through B and E is:

m = (2.5 - 2) / (-0.5 - 9) = 0.25

The midpoint of BE is:

N = ((9-0.5)/2, (2+4)/2) = (4.25, 3)

Therefore, the equation of median BE is:

y - 3 = 0.25(x - 4.25)

Simplifying, we get:

y = 0.25x + 1.375

The equation of median CF can also be found in a similar way. The slope of the line passing through C and F is:

m = (3 - 1) / (2 - 4) = -1

The midpoint of CF is:

P = ((-5+2)/2, (4+3)/2) = (-1.5, 3.5)

Therefore, the equation of median CF is:

y - 3.5 = -1(x + 1.5)

Simplifying, we get:

y = -x + 2

(b) To show that the three medians all 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 intersection of medians AD and BE. To do this, we can solve the system of equations:

y = -0.25x + 3

y = 0.25x + 1.375

Substituting the second equation into the first, we get:

0.25x + 1.375 = -0.25x + 3

0.5x = 1.625

x = 3.25

Substituting this value of x into either equation, we get:

y = -0.25(3.25) + 3 = 2.375

Therefore, the intersection of medians AD and BE is the point (3.25, 2.375).

Now let's check if median CF also passes through this point. Substituting x = 3.25 into the equation of median CF, we get:

y = -3.25 + 2 = -1.25

Therefore, the point (3.25, 2.375) is not on median CF.

However, this does not mean that the three medians do not intersect at the same point. In fact, they always do! This point of intersection is called the centroid of the triangle, and it is the average of the three vertices. In this case, the centroid is:

G = ((-5+9+4)/3, (4+2+1)/3) = (2.67, 2.33)

So all three medians pass through the point (2.67, 2.33), which is the centroid of triangle ABC.

Mar 5, 2023