Note: The distance AB is obtained from the difference of the x-coordinates of the two points. Find the equation of the line which passes through the point (1, 3) and is perpendicular to the line whose equation is y = 2x + 1. To introduce the idea, consider the grid above. Prove that the diagonals of a rhombus bisect each other at right angles using coordinate geometry. The distance between the points (1, a) and (4, 8) is 5. If two lines l1 and l2 are parallel then corresponding angles are equal. The two lines have the same gradient and so are parallel. If the lines are parallel then PQX = ABY (corresponding angles). Find the distance between the points A(−4, −3) and B(5, 7). After Descartes invention of the coordinate system, other systems followed. Coordinate geometry can be used to prove results in Euclidean Geometry. The vertical number line is the y- axis The origin is where the two intersect. Plot the two points and draw the line through them. Conversely If the product is −1, then AB = PQ. (These A line passes through the points (1, 2) and (5, 10). There are three ways commonly used in coordinate geometry: Give the coordinates of any two points on the line; Give the coordinates of one point on the line, and the slope of the line; Give an equation that defines the line. The second diagram shows a line parallel to the x-axis and it has a y-intercept at C. The third diagram shows a line parallel to the y-axis and it has an x-intercept at D. When we plot points which satisfy the equation y = 2x + 1 we find that they lie in a straight line. It is important that the reader be able to not only use algebraic processes but also to understand them fully, if he is to apply them with confidence. The equation of every line can be put in general form. If we were to place a point on the plane, coordinate geometry gives us a way to … It does not matter whether we are talking about a line, ray or line segment. Similarly the gradient of BA = − which is the same as the gradient of AB. Form the right-angled triangle PQX, where X is the point (x2, y1), PX = x2 − x1 or x1 − x2 and QX = y2 − y1 or y1 − y2. This is proved below. Define the equations of curves, circles and ellipses. For example, the equation x2 + y2 = 1, describes a circle of unit radius in the plane. Prove that the lines joining the midpoints of opposite sides of a quadrilateral and the lines joining the midpoints of its diagonals meet in a point and bisect each other. Conversely, if corresponding angles are equal then the lines are parallel. The results and ideas of the ancient Greeks provided a motivation for the development of coordinate geometry. It is important that the reader be able to not only use algebraic processes but also to understand them fully, if … Thus the product of the gradients of the diagonals = −1. The point with coordinates (4, 2) has been plotted on the Cartesian plane shown. An important aspect of doing this is placing objects on the Cartesian plane in a way that minimises calculations. Rewrite the equation 2x + 3y = 6 in the form y = mx + c and hence find the value of the gradient and y-intercept. What is meant by Coordinate Geometry? Conversely. A line passes through the point (5, 7) and has gradient . Coordinate geometry has traditionally been attributed to René Descartes (1599−1650) and Pierre de Fermat (1601−1665) who independently provided the beginning of the subject as we know it today. This will make more sense as we proceed. Hence the x-coordinate of M is the average of x1 and x2, and y-coordinate of M is the average of y1 and y2. M = N and so the midpoints coincided which means that the diagonals bisect each other. Find the gradient of the line passing through (a, b) and (0, c). Show that the line through the points A(6, 0) and B(0, 12) is perpendicular to the line through P(8, 10) and Q(4, 8). about them. Find the coordinates of the midpoint of the line interval joining the points (6, 8) and (−3, 2). the "x axis" and another a right angles to it called the y axis. What is Coordinate Geometry? It brought together nearly all of algebra and geometry using the coordinate plane. Note that since = it does not matter which point we take as the first and which point we take as the second. Notice that as you move from A to B along the interval the y-value increases as the x-value increases. Write down the gradient and y-intercept of the line with equation y = 3x − 4. Thus the coordinates of the midpoint M are (3, 5). Using the y-intercept and one other point. The coordinate plane or Cartesian plane is a basic concept for coordinate geometry. In particular it is central to the mathematics students meet at school. b The midpoint of AB has coordinates 1, . If the gradients are equal = . Note that it is usual to give the answer in the form y = mx + c, Equation of a straight line given two points. After the invention of the rectangular coordinate system, algebraic representations became available to points, lines, curves, and other geometric constructions and drawing. The general form is not unique. Using the y-intercept and a second point the equation can be found . The gradient is positive. Suppose that P(x1, y1) and Q(x2, y2) are two points. That is, ‘ If the product of the gradients of two lines is −1 then they are perpendicular’. Let the coordinates of the vertices be O(0, 0), A(a, 0), B(a + c, d) and C(c, d). In any triangle ABC prove that AB2 + AC2 = 2(AD2 + DC2). Recall that a plane is a flat surface that goes on forever in both directions. If the lines are perpendicular, POQ = AOB. The equations y = 2x − 3, x = 6 and 2x − 3y = 6 can be written as −2x + y + 3, x − 6 = 0 and 2x − 3y − 6 = 0 respectively. This method does not work if the line is parallel to an axis or passes through the origin. A square has vertices O(0, 0), A(a, 0), B(a, a) and C(0, a). 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