If the interval is vertical, the run is zero Matter which point we take as the first and Similarly the gradient of BA = − which is the same as the gradient of AB. Notice that in this case as we move from A to B the y value decreases as the x value increases. Notice that as you move from A to B along the interval the y-value increases as the x-value increases. We will usually the pronumeral m for gradient. In coordinate geometry the standard way to define the gradient of an interval AB is where rise is the change in the y-values as you move from A to B and run is the change in the x-values as you move from A to B. There are several ways to measure steepness. 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 If the line is going down, from left to right, the slope is negative.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).If the line is going up, from left to right, the slope is positive.You can double check your answer by trying different points on the line.You can pick any two points on a line to calculate the slope.Slope = change in y over the change in x.The run is the distance that the line travels from left to right. The rise is the distance that the line travels up or down. You can draw a right triangle using any two points on the line. You can see this on the two example problems above.Īnother way to remember how the slope works is "rise over run". If you look at the line from left to right, a line that is moving up will have a positive slope and a line that is moving down will have a negative slope. Since you can't divide by 0, a vertical line has an undefined slope. The change in y is 0, so the slope is 0.Ī vertical line has a change in x of 0. Some special cases include horizontal and vertical lines.Ī horizontal line is flat. You can see that the line contains the points (-2,4) and (2, -2).
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