The coordinates of triangle ABC areA(0,0), B(2,6), and C(4,2). Using coordinates geometry; prove that, if the midpoints of sides AB and AC are joined, the segment formed is parallel to the third side and equal to one- half the length of the third side.

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To prove that the segment formed by joining the midpoints of sides AB and AC is parallel to side BC and equal to one-half the length of side BC, we can use slope and distance formulas in coordinate geometry.

Let's begin by finding the midpoints of sides AB and AC.

Midpoint of AB:
The coordinates of A are (0,0) and the coordinates of B are (2,6). To find the midpoint, we average the x-coordinates and the y-coordinates of A and B separately.

Midpoint of AB = ((0+2)/2, (0+6)/2) = (1, 3)

Midpoint of AC:
The coordinates of A are (0,0) and the coordinates of C are (4,2). To find the midpoint, we average the x-coordinates and the y-coordinates of A and C separately.

Midpoint of AC = ((0+4)/2, (0+2)/2) = (2, 1)

Now, let's find the slope of side BC and the segment connecting the midpoints of AB and AC.

Slope of side BC (mBC):
The coordinates of B are (2,6) and the coordinates of C are (4,2). The slope formula is given by (y2 - y1)/(x2 - x1).

mBC = (2-6)/(4-2) = -4/2 = -2

Slope of the segment joining midpoints (mMid):
The coordinates of midpoint of AB are (1,3) and the coordinates of midpoint of AC are (2,1). The slope formula is given by (y2 - y1)/(x2 - x1).

mMid = (1-3)/(2-1) = -2/1 = -2

Since both mBC = -2 and mMid = -2, we can conclude that the segment joining the midpoints is parallel to side BC.

Now, let's find the length of side BC and the length of the segment connecting the midpoints.

Length of side BC:
The coordinates of B are (2,6) and the coordinates of C are (4,2). The distance formula is given by sqrt((x2 - x1)^2 + (y2 - y1)^2).

Length of BC = sqrt((4-2)^2 + (2-6)^2) = sqrt(2^2 + (-4)^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5)

Length of the segment connecting midpoints:
The coordinates of midpoint of AB are (1,3) and the coordinates of midpoint of AC are (2,1). The distance formula is given by sqrt((x2 - x1)^2 + (y2 - y1)^2).

Length of segment connecting midpoints = sqrt((2-1)^2 + (1-3)^2) = sqrt(1^2 + (-2)^2) = sqrt(1 + 4) = sqrt(5)

We can observe that the length of the segment connecting the midpoints is equal to one-half the length of side BC, as shown by:

sqrt(5) = (2sqrt(5))/2 = sqrt(5)

Hence, we have proved that the segment formed by joining the midpoints of sides AB and AC is parallel to side BC and equal to one-half the length of side BC, using coordinates geometry.