The gradient of the line y = −2 x + 6 is −2. In general, the equation of the horizontal line through P( a, b) is y = b.įind the equation of the line that is parallel to the line y = −2 x + 6 and passing through the point A(1, 10). The equation of the horizontal line through the point (9, 5) is y = 5. The equation of the horizontal line through the point (0, 5) is y = 5. In a horizontal line all points have the same y-coordinate, but the x-coordinate can take any value. In general, the equation of the vertical line through P( a, b) is x = a.īecause this line does not have a gradient it cannot be written in the formĪ horizontal line has gradient 0. All the points on this line have x-coordinate 6. The equation of the vertical line through the point (6, 0) is x = 6. In a vertical line all points have the same x-coordinate, but the y-coordinate can take any value. Conversely, the points whose coordinates satisfy the equation y = mx + c always lie on the line with gradient m and y-intercept c. That is, the line in the cartesian plane with gradient m and y-intercept c has equation That is, the intervals have the same gradient.Ĭonsider the line with gradient m and y-intercept c. Triangle ABX is similar to triangle PQY since the corresponding angles are equal. Draw right-angled triangles ABX and PQY with sides AX and PY parallel to the x-axis and sides BX and QY parallel to the y-axis. Suppose AB and PQ are two intervals on the same straight line. This definition depends on the fact that two intervals on a line have the same gradient. The gradient of a line is defined to be the gradient of any interval within the line. If the interval is horizontal, the rise is zero 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 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)
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