The gradient is a measure of the steepness of line. Take the average of the x-coordinates and the average of the y-coordinates. The midpoint of an interval with endpoints P( x 1, y 1) and Q( x 2, y 2) is. Hence the x-coordinate of M is the average of x 1 and x 2, and y-coordinate of M is the average of y 1 and y 2. Triangles PMS and MQT are congruent triangles (AAS), and so PS = MT and MS = QT. Suppose that P( x 1, y 1) and Q( x 2, y 2)are two points and let M( x, y) be the midpoint. We can find a formula for the midpoint of any interval. Thus the coordinates of the midpoint M are (3, 5). The y coordinate of M is the average of 2 and 8. Hence the x-coordinate of M is the average of 1 and 5. Triangles AMS and MBT are congruent triangles (AAS), and so AS = MT and MS = BT. When the interval is not parallel to one of the axes we take the average of the x-coordinate and the y-coordinate. Note: 4 is the average of 1 and 7, that is, 4 =. Midpoint is at (4, 2), since 4 is halfway Note that ( x 2 − x 1) 2 is the same as ( x 1 − x 1) 2 and therefore it doesn’t matter whether we go from P to Q or from Q to P − the result is the same.įind the coordinates of the midpoint of the line interval AB, given:Ī A(1, 2) and B(7, 2) b A(1, −2) and B(1, 3) PX = x 2 − x 1 or x 1 − x 2 and QX = y 2 − y 1 or y 1 − y 2 Suppose that P( x 1, y 1) and Q( x 2, y 2) are two points.įorm the right-angled triangle PQX, where X is the point ( x 2, y 1), We can obtain a formula for the length of any interval. The distance between the points A(1, 2) and B(4, 6) is calculated below. Pythagoras’ theorem is used to calculate the distance between two points when the line interval between them is neither vertical nor horizontal. The example above considered the special cases when the line interval AB is either horizontal or vertical. The difference of the y-coordinates of the Swap the x coordinate of C with the y-coordinate of D.Find the distance between the following pairs of points.Ī A(1, 2) and B(4, 2) b A(1, −2) and B(1, 3) Swap the y-coordinate of B with the x-coordinate of D. Swap the x-coordinate of B with the y-coordinate of D. Swap the x-coordinate of B with the y-coordinate of C. Swap the y-coordinate of A with the x-coordinate of D. Swap the x-coordinate of A with the y-coordinate of B. You can also swap coordinates in the following 9 ways: You can always swap the y-coordinates of the points in the first and second quadrant (x2), the y-coordinates of the points in the first and third quadrant (x2), both (x2), swap the x-coordinates of the points in the first and fourth quadrant (x2), swap the x-coordinates of the points in the second and third quadrant (x2), or – again – both (x2), to get another solution. However, since you need 8 coordinates and 0 can’t be any of them, your points will always end up with the same 8 numbers as coordinates:
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