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Reference: Cartesian Coordinates.
to the axis of symmetry. An axis of symmetry is sometimes called a mirror line or line of symmetry.
Some shapes have more than one axis of symmetry, like the square (figure b). Other shapes have no axes of symmetry, like the parallelogram (figure c).
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To find more information on the symmetry properties of various shapes, search for them under the respective entries.
References: Image, Mediator, Mirror Line, Object, Rotational Symmetry.
BALANCING AN EQUATION
This is a method of solving equations in which the equals sign is the point of balance of the equation. To begin with, the expression on the left-hand side of the equals sign is equal to the expression on the right-hand side. The balance of the two sides of the equation is maintained by performing the same operation on each side. The process of applying these operations leads to solving the equation. The operations are adding, subtracting, multiplying, dividing, squaring, and taking the square root.
Example. Solve the equation 2x — 3 = — 8 + a:.
Solution. The left-hand side of the equation is 2x — 3, which is balanced by the right hand side of the equation, which is —8 + x. To solve this equation, we need to finish up with x on left-hand side of the equation and a number called the solution on the right-hand side. Whatever operation is done on one side of the equation must, at the same time, be done on the other side of the equation, thus “balancing the equation.” This technique of solving by balancing the equation is demonstrated in this example. Write
2x — 3 = — 8 + x 2x — 3 — x = — 8 + a; — x Subtracting x from both sides
x — 3 = — 8 Simplifying, because 2x — x = x and +x — x = 0
x — 3 + 3 = —8 + 3 Adding 3 to both sides
x = — 5 Simplifying because —3 + 3 = 0 and — 8 + 3 = — 5
The solution of the equation is a: = — 5.
For setting out purposes, we ensure that there is only one equals sign per line of working, and the equals signs are kept in a vertical line directly underneath each other.
References: Equations, Inverse Operation, Linear Equation, Quadratic Equations, Solving an Equation.
BAR GRAPH 49
Reference: Bar graph.
The bar graph, also called a bar chart or column graph, is used in statistics to display data. A bar graph consists of a number of vertical bars, or rectangles, of equal widths whose heights are proportional to the frequencies of certain quantities. If the data are discrete, or in separate categories, a small space is left between each bar. For continuous data a histogram is used. Data which may be contained in a frequency table are displayed much more effectively in a bar graph so that comparisons of quantities can be made more easily and with a greater impact. Whenever a bar graph is drawn, make sure it has a title, the axes are numbered, or named, and labeled, and the heights of the columns represent the frequencies.
Example. The frequency table shows how the 20 people in Peter’s city office travel to work. Draw a bar graph to display these data.
Method of travel Bus Car Cycle Train Walk
Frequency (f) 6 2 7 1 4
Solution. Columns in the bar graph represent each of the methods of travel to work, and the heights of the columns represent their frequencies (see figure). The graph is given a suitable title. The frequency axis is numbered, and the methods of transport are labeled on the horizontal axis. The heights of the columns represent the frequencies.
If desired, the bars can be drawn horizontally instead of vertically.
Method of Transport to Work for Office Workers
References: Bar Chart, Column Graph, Continuous Data, Data, Discrete Data, Distribution, Frequency, Frequency Table.
50 BASE (GEOMETRY)
When a polygon is drawn, the base is the side at the bottom of the shape. The polygon can be rotated so that another of its sides is the base. For example, suppose the polygon is a triangle. Each side of a triangle in turn can be the base, depending on how the triangle is drawn. The same triangle is drawn in figure a in three different positions showing how each side in turn is the base of the triangle. The altitude of a triangle is its perpendicular height, which is the shortest distance from its highest point to its base.
In the first position of the triangle in figure a the side c is the base, in the second position side a is the base, and in the third position side b is the base. When calculating the area of a triangle it may be advantageous to carefully select one side as the base in preference to the others. This point is illustrated in the following example.
Example. Calculate the area of the right-angled triangle ABC in figure b. The measurements are in meters.
Solution. In order to find the area of the right-angled triangle ABC, we will use AC as the base of the triangle and AB as the height.