ABC has verticals A (0, 0), B (2,2), and C (5,-1). Find the coordinates of L, the midpoint of AC, and M, the midpoint of BC. Verify that LM ll AB and LM = ½ AB
To find the coordinates of L, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by the average of the x-coordinates and the average of the y-coordinates.
For AC, the coordinates of point A are (0, 0) and the coordinates of point C are (5, -1). Therefore, the x-coordinate of L is (0 + 5) / 2 = 2.5 and the y-coordinate of L is (0 - 1) / 2 = -0.5. So, the coordinates of L are (2.5, -0.5).
To find the coordinates of M, we can use the same midpoint formula. The coordinates of point B are (2, 2) and the coordinates of point C are (5, -1). Therefore, the x-coordinate of M is (2 + 5) / 2 = 3.5 and the y-coordinate of M is (2 - 1) / 2 = 0.5. So, the coordinates of M are (3.5, 0.5).
Now, let's verify that LM is parallel to AB. We'll calculate the slope of both lines. The slope of AB can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) = (0, 0) and (x2, y2) = (2, 2). The slope of AB is (2 - 0) / (2 - 0) = 2/2 = 1.
Similarly, the slope of LM can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) = (2.5, -0.5) and (x2, y2) = (3.5, 0.5). The slope of LM is (0.5 - (-0.5)) / (3.5 - 2.5) = 1 / 1 = 1.
Since the slopes of AB and LM are equal, we can conclude that LM is parallel to AB.
Finally, let's calculate the length of both LM and AB. The distance between two points (x1, y1) and (x2, y2) is given by the formula sqrt((x2 - x1)^2 + (y2 - y1)^2).
The length of AB between (0, 0) and (2, 2) is sqrt((2 - 0)^2 + (2 - 0)^2) = sqrt(4 + 4) = sqrt(8).
The length of LM between (2.5, -0.5) and (3.5, 0.5) is sqrt((3.5 - 2.5)^2 + (0.5 - (-0.5))^2) = sqrt(1 + 1) = sqrt(2).
Since LM = sqrt(2) and AB = sqrt(8), we can see that LM = 1/2 * AB.
Therefore, we have verified that LM is parallel to AB and that LM = 1/2 * AB.