C1: The Distance Between Two Points & The Mid-Point of a Line Learning Objectives : to be able to apply Pythagoras’ Theorem to calculate the distance between two points and to be able to use mean averages to calculate the half way point on a line

Starter: Calculate the length of the side of a right angled triangle with hypotenuse 8cm and shorter side 5.5cm. Find the equation of the line joining points (-4, -1) and ( 3, 9).

The Cartesian coordinate system The Cartesian coordinate system is named after the French mathematician René Descartes (1596 – 1650). Points in the (x, y) plane are defined by their perpendicular distance from the x- and y-axes relative to the origin, O. The coordinates of a point P are written in the form P(x, y). The x-coordinate, or abscissa, tells us the horizontal distance from the y-axis to the point. Introduce the system of Cartesian coordinates as providing us with a way to express the geometry of lines, curves and shapes algebraically. Explain that points are normally denoted by a capital letter followed by the coordinates of the point. Use the embedded Flash movie to demonstrate a variety of points in each of the four quadrants and on the axes. Observe that the x-coordinate of the point becomes negative as the point passes to the left of the y-axis and that the y-coordinate becomes negative as it passes below the x-axis. The y-coordinate, or ordinate, tells us the vertical distance from the x-axis to the point.

The distance between two points Given the coordinates of two points, A and B, we can find the distance between them by adding a third point, C, to form a right-angled triangle. We then use Pythagoras’ theorem. This method for finding the shortest distance between two given points is the same as finding the length of a line segments joining the two points.

A General Formula: What is the distance between two general points with coordinates A(x1, y1) and B(x2, y2)? The horizontal distance between the points is . x2 – x1 The vertical distance between the points is . y2 – y1 Using Pythagoras’ Theorem, the square of the distance between the points A(x1, y1) and B(x2, y2) is This slide shows the generalization for finding the distance between two points. The distance between the points A(x1, y1) and B(x2, y2) is

What is the distance between the points A(5, –1) and B(–4, 5)? Example Given the coordinates of two points we can use the formula to directly find the distance between them. For example: What is the distance between the points A(5, –1) and B(–4, 5)? x1 y1 x2 y2 A(5, –1) B(–4, 5) Point out that it doesn’t matter which point is called (x1, y1) and which point is called (x2, y2). It can help to write x1, y1, x2 and y2 above each coordinate as shown before substituting the values into the formula. The answer in this example is written in surd form. An alternative would be to write it to a given number of decimal places; for example, 10.82 (to two decimal places).

Task 1 : Calculate the distance between the following pairs of points Task 1 : Calculate the distance between the following pairs of points. Give your answers to 3s.f. (3, 7) (9, 4) (3, -5) (6, -3) (-5, -2) (-7, -6) (-3, 4) (5, 4) (-3.5, 4.7) (7.5, 1.3) (2.5, -3.9) (-8.7, 0.9)

The Mid-Point of a Line In general, the coordinates of the mid-point of the line segment joining (x1, y1) and (x2, y2) are given by: (x2, y2) (x1, y1) x y is the mean of the x-coordinates. Talk through the generalization of the result for any two points (x1, y1) and (x2, y2). As for the generalization for the distance between two points, it doesn’t matter which point is called (x1, y1) and which point is called (x2, y2). is the mean of the y-coordinates.

Example: Use this activity to explore the mid-points of given line segments. Establish that the x-coordinate of the mid-point will be half way between the x-coordinates of the end points. This is the mean of the x-coordinates of the end-points. The y-coordinate of the mid-point will be half way between the y-coordinates of the end points. This is the mean of the y-coordinates of the end-points.

Task 2 : Find the mid- point of each line segment (3, 7) (9, 4) (3, -5) (6, -3) (-5, -2) (-7, -6) (-3, 4) (5, 4) (-3.5, 4.7) (7.5, 1.3) (2.5, -3.9) (-8.7, 0.9)

Task 3 Find the length of the line from the origin to the point (7, 4). Show, using Pythagoras’ Theorem, that the lines joining A(1, 6), B(-1, 4) and C(2, 1) form a right angled triangle. Show that ΔABC is isosceles where A, B and C are the points (7, 3), (-4, 1) and (-3, -2). Then, by finding the mid point of the base, calculate the area of the triangle.