The midpoint of the segment joining (a,b) and (3a,-3b) has coordinate (2a,-b). Is this statement true or false? If it is true, then how?

Question

The midpoint of the segment joining (a,b) and (3a,-3b) has coordinate (2a,-b). Is this statement true or false? If it is true, then how?

Answer 1

The coordinates of the midpoint between two known points can be though of as the average of the two known points. Given the known points (a,b) and (3a,-3b), we can find the coordinates of the midpoint by taking the average of the x and y coordinates of the two given points. in the first point, the x coordinate is a, and in the second point, the x coordinate is 3a, so to take the average of the x coordinates is equal to (a + 3a) / 2, which simplifies to 4a/2, or 2a, so the x coordinate of the midpoint is 2a. We can repeat this process for the y coordinates. If we take the y coordinate of the first point, b, and the y coordinate of the second point, -3b, we can say that the average of them is given by (b - 3b) / 2, which is equal to -2b / 2, or -b, which we know is the y coordinate of the midpoint of the given points.

Therefore, the midpoint of (a,b) and (3a,-3b) is (2a, -b), and we can see that the statement is true.

Answer 2

The mid-point is ((a + 3a)/2, (b - 3b)/2)