Geometry Revision Notes

The Coordinate Plane

The coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They intersect at the origin (0, 0). The plane is divided into four quadrants by these axes. Points in the first quadrant have positive x and positive y coordinates (+, +). The second quadrant has negative x and positive y (-, +). The third quadrant has negative x and negative y (-, -). The fourth quadrant has positive x and negative y (+, -).

Plotting Points using (x, y)

To plot a point given its coordinates (x, y), start at the origin (0, 0). The first number, x, tells you how far to move horizontally. Move right for positive x, and left for negative x. The second number, y, tells you how far to move vertically. Move up for positive y, and down for negative y. Always remember the order: 'x' comes before 'y', just like in the alphabet. For instance, to plot (3, -2), move 3 units right, then 2 units down.

Identifying Coordinates of a Plotted Point

To find the coordinates of a point already plotted on a grid, start by moving horizontally from the point to the x-axis to find its x-value. Then, move vertically from the point to the y-axis to find its y-value. Write these as an ordered pair (x, y). For example, if a point is 4 units left of the y-axis and 1 unit above the x-axis, its coordinates are (-4, 1).

The Coordinate Plane

The coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They intersect at the origin (0, 0). This creates four quadrants, labeled with Roman numerals starting from the top-right and moving anticlockwise. Points in Quadrant I have both x and y positive (+,+), Quadrant II have x negative and y positive (-,+), Quadrant III have both x and y negative (-,-), and Quadrant IV have x positive and y negative (+,-). Remember that points on the axes are not considered to be in any quadrant.

Plotting Points (x, y)

To plot a point (x, y) on the coordinate plane, always start at the origin (0, 0). The first number, 'x', tells you the horizontal movement. A positive 'x' means move right, and a negative 'x' means move left. The second number, 'y', tells you the vertical movement. A positive 'y' means move up, and a negative 'y' means move down. For example, to plot (4, -2), move 4 units right from the origin, then 2 units down. Always list the x-coordinate first, then the y-coordinate. This order is crucial for accuracy.

Reading Coordinates from a Graph

To read the coordinates of a point already plotted on a graph, first determine its horizontal position relative to the origin. This is the x-coordinate. Look directly up or down from the point to see which value it aligns with on the x-axis. Then, determine its vertical position. This is the y-coordinate. Look directly left or right from the point to see which value it aligns with on the y-axis. Write the coordinates as an ordered pair (x, y). For example, a point 3 units to the left of the y-axis and 5 units above the x-axis would have coordinates (-3, 5).

Distance Between Two Points

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be calculated using the distance formula, which is derived from the Pythagorean theorem: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$. It's important to be careful with signs when subtracting coordinates. For example, the distance between (1, 2) and (4, 6) is $\sqrt{((4 - 1)^2 + (6 - 2)^2)} = \sqrt{(3^2 + 4^2)} = \sqrt{(9 + 16)} = \sqrt{25} = 5$ units. This formula is fundamental for proving properties of shapes, such as the lengths of sides in a polygon.

Midpoint of a Line Segment

To find the midpoint M of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, use the midpoint formula: $M = ((\frac{x_1 + x_2}{2}), (\frac{y_1 + y_2}{2}))$. This formula essentially finds the average of the x-coordinates and the average of the y-coordinates. For example, the midpoint of the segment connecting (1, 2) and (5, 8) is $((\frac{1 + 5}{2}), (\frac{2 + 8}{2})) = (\frac{6}{2}, \frac{10}{2}) = (3, 5)$. This is useful for finding the centre of a shape or for bisection problems.

Gradient of a Line

The gradient (or slope) of a line connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $m = (\frac{y_2 - y_1}{x_2 - x_1})$. A positive gradient indicates an upward slope from left to right, while a negative gradient indicates a downward slope. A larger absolute value of the gradient means a steeper line. For example, the gradient between (2, 3) and (6, 11) is $m = (\frac{11 - 3}{6 - 2}) = (\frac{8}{4}) = 2$. Two lines are parallel if they have the same gradient. Two lines are perpendicular if the product of their gradients is -1 (unless one is horizontal and the other vertical).

Applications in Geometric Proofs

Higher methods in coordinate geometry are often used to prove properties of geometric figures. For instance, to prove a quadrilateral is a square, you would use the distance formula to show all sides are equal, and the gradient formula to show adjacent sides are perpendicular. To show a triangle is isosceles, you would show two sides have equal lengths. For a right-angled triangle, you could show that the gradients of two sides satisfy the perpendicular condition. For a parallelogram, you could show opposite sides have the same gradient (are parallel) or that diagonals bisect each other (same midpoint). Always clearly state which formula you are using and what property you are trying to demonstrate.

What is a Perpendicular Bisector?

A perpendicular bisector is a line that divides a line segment into two equal parts and is also at a 90-degree angle to the original line segment. It passes through the midpoint of the line segment. This means that if you have a line segment AB, its perpendicular bisector will pass through the midpoint M of AB, and the line formed will be perpendicular to AB. Any point on the perpendicular bisector is equidistant from points A and B.

Constructing a Perpendicular Bisector with Compass and Straightedge

To construct a perpendicular bisector of a line segment AB: (1) Place the compass point on point A and open it to more than half the length of AB. Draw an arc above and below the line segment. (2) Without changing the compass width, place the compass point on point B and draw another arc above and below the line segment, ensuring it intersects the first set of arcs. (3) You will now have two intersection points. Use a straightedge to draw a straight line connecting these two intersection points. This line is the perpendicular bisector of AB.

Properties and Uses of a Perpendicular Bisector

The key properties are: it bisects (cuts in half) the line segment, and it is perpendicular (at 90 degrees) to the line segment. Every point on the perpendicular bisector is equidistant from the two endpoints of the original line segment. Perpendicular bisectors are used in various geometric constructions, such as finding the circumcentre of a triangle (the centre of a circle that passes through all three vertices of the triangle, found at the intersection of the triangle's perpendicular bisectors).