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.
Hint:
Parallel means they have the same slope.
Slope BC = (yc-yb)/(xc-xb)=-2
Let H be mid-point of AB = (1,3)
Let K be mid-point of AC = (2,1)
Slope of HK = (yk-yh)/(xk-xh)=-2/1=-2
Therefore HK is parallel to BC.
To prove that the segment joining the midpoints of sides AB and AC is parallel to side BC and equal to half the length of side BC, we can use coordinate geometry.
Let's begin by finding the coordinates of the midpoints of sides AB and AC.
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) can be found by using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Using this formula, we can find the midpoints of sides AB and AC as follows:
Midpoint of AB = ((0 + 2) / 2, (0 + 6) / 2) = (1, 3)
Midpoint of AC = ((0 + 4) / 2, (0 + 2) / 2) = (2, 1)
Now, let's find the equation of the line passing through the midpoints of AB and AC. We will use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
We already know two points on the line: (1, 3) and (2, 1). Let's find the slope of the line using these points:
m = (y2 - y1) / (x2 - x1)
= (1 - 3) / (2 - 1)
= -2 / 1
= -2
Now we have the slope of the line passing through the midpoints of AB and AC. Let's use this slope and one of the midpoints (let's use (1, 3)) to find the equation of the line:
y - 3 = -2(x - 1)
Expanding this equation, we get:
y - 3 = -2x + 2
Simplifying further:
y = -2x + 5
Now, let's find the equation of side BC. We can use the coordinates of points B and C, which are (2, 6) and (4, 2) respectively.
The slope of side BC can be found using these points:
m = (y2 - y1) / (x2 - x1)
= (2 - 6) / (4 - 2)
= -4 / 2
= -2
Using the point-slope form of a linear equation with point B(2, 6), we can find the equation of side BC:
y - 6 = -2(x - 2)
Expanding this equation, we get:
y - 6 = -2x + 4
Simplifying further:
y = -2x + 10
From the equations of the lines passing through the midpoints of AB and AC (y = -2x + 5) and side BC (y = -2x + 10), we can see that they have the same slope (-2). This indicates that the lines are parallel.
Furthermore, we can calculate the lengths of side BC and the segment joining the midpoints (AB and AC) to verify that they are equal.
The length of a line segment with endpoints (x1, y1) and (x2, y2) can be found using the distance formula:
Length = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can find the lengths of side BC and the segment joining the midpoints:
Length of BC = √((4 - 2)^2 + (2 - 6)^2) = √(2^2 + (-4)^2) = √(4 + 16) = √20 = 2√5
Length of the segment joining the midpoints = √((2 - 1)^2 + (1 - 3)^2) = √(1^2 + (-2)^2) = √(1 + 4) = √5
As we can see, the lengths of side BC and the segment joining the midpoints are equal.
Therefore, we have proven that the segment joining the midpoints of sides AB and AC is parallel to side BC and equal to half the length of side BC using coordinate geometry.