REVIEW – PROPORTIONAL REASONING
A ratio is a comparison between two numbers measured in the same units.
A ratio can be expressed in three ways as shown below:
Ratios, like fractions, can be simplified. For example, the ratio 150 : 15 can also be expressed
Notice that the numerator of the fraction is larger than the denominator. This can be common with ratios.
If two ratios are equivalent (equal), the first (top) term of each ratio compares to the second (bottom) term in an identical manner. You can represent this equivalence in the two ratios here:
An equation showing equivalent ratios is called a proportion.
Cross Multiply and Divide
When two fractions are equal to each other, any unknown numerator or denominator can be found. The following example shows the process.
Solution: Cross multiply means multiply the numbers across the equals sign (the arrow). The divide part means divide that result by the number opposite the unknown ( x ) as shown below.
This gives the result x = 3 × 2.1 ÷ 4
In other words, if
Two figures are said to be similar figures if they have the same shape but are different sizes. A diagram drawn to scale to another diagram makes two similar figures. Also, an enlargement or a reduction of a photograph when reproduced to scale, produces similar figures.
Corresponding angles are two angles that occupy the same relative position on similar figures. Corresponding sides are two sides that occupy the same relative position in similar figures. When we use the term “relative position,” you must remember that the one figure might be turned compared to the other figure. It is necessary to look arrange the two figures so they look the same before deciding which angles or sides correspond.
When labelling figures, strings of capital letters in alphabetical order are used. The order of the letters tells you which sides and angles correspond.
The two quadrilaterals are similar. Because ABCS is similar to WXYZ, we can use a symbol “~” which means “is similar to.” So ABCD ~ WXYZ
DETERMINING SIDES IN SIMILAR FIGURES
When working with the length of sides in similar figures, because the figures are always a reduction of enlargement of each other, the ratio of the corresponding sides is always the same. What this means is that by using a proportion, you can determine the lengths of all the sides in both figures.
Example 1: The two figures below are similar. Find the lengths of the side of the smaller figure.
Set up proportions using BC and GH as those two sides define the ratio. For this example, make sure the sides from the big figure are always on the top and the sides for the small figure are always on the bottom.
The lengths of the smaller figure are: FG = 6 in., HI = 7 in., IJ = 4 in. And JF = 3 in.
DETERMINING ANGLES IN SIMILAR FIGURES
Since corresponding angles in similar figures must be equal, the only difficulty with determining the angle measures is making sure that the figures are arranged so they look the same. Sometimes this will already be done for you. But other times, you must carefully look at this arrangement.
Example: If ΔRST is similar to ΔLMN, and the angle measure for ΔLMN are as listed below, what are the angle measure for the angles in ΔRST?
Solution: Determine which angles correspond, and those angle measures are equal.
ASK YOUR TEACHER FOR UNIT QUIZ 1.
SCALE FACTOR IN SIMILAR FIGURES
When figures are enlarged or reduced, this is often done by a scale factor. A scale factor is the ratio of a side in one figure compared to the corresponding side in the other figure. Earlier in this unit, we used the ratio of two corresponding sides in a proportion to calculate other sides. The difference with using a scale factor is the ratio when using scale factor is that it is always compared to 1. So a proportion is not necessary when the scale factor is 1: some number, e.g. 1:500.
Usually the scale factor is a single number: example, the scale factor is 1.5 or the scale factor is one quarter. Whether dealing with an enlargement or a reduction, the process of solving the problem is the same. To solve this, multiply the original lengths by the scale factor to produce the scaled lengths.
Example 1: A tissue has the dimensions of 9 cm by 10 cm. The company that makes the tissues wants to increase the dimensions of the tissues by 1.7. What are the new dimensions of the tissues?
Solution: To get the new size, multiply each dimension by 1.7.
length: 10 cm × 1.7 = 17 cm
width: 9 cm × 1.7 = 15.3 cm
Scale factors are also used on maps where a unit on a map represents a certain actual distance on the ground. For example, a scale factor might be 1 cm represents 5 km.
Example 2: The scale on a neighbourhood map shows that 1 cm on the map represents an actual distance of 2.5 km.
a) On the map, Waltham Street has a length of 14 cm. What would
the actual length of street be?
b) Centre Street has an actual length of 25 km. What would the length
of the street be on the map?
a) Multiply the map length by the scale factor.
14 cm × 2.5 = 35 km
b) Divide the actual distance by the scale factor.
25 km ÷ 2.5 = 10 cm
Proportions can also be used, including the English words and numbers, as before.
CALCULATING SCALE FACTOR
In the previous section, we used a given scale factor to calculate the length of sides when a figure is enlarged or reduced. In this section, we will learn about calculating the scale factor when the two corresponding sides in similar figures are given.
Use a proportion to determine the scale factor. Remember, a scale factor is always 1:x where x is the number we are looking for. It may be stated as just a number, but it is really a ratio.
Adam is drawing a scale drawing of a staircase. On the drawing, the height of one stair is 0.5 cm while the actual height of the stair is 20 cm. What was the scale factor that Adam used?
Set up a ratio and divide to calculate the scale factor.
Scale Factor = x = 20 × 1 ÷ 0.5 = 40
It is also important to note that when calculating scale factor, the units of the two numbers MUST be the same. You cannot calculate scale factor with cm and metres, for example. You must change one unit into the other before using the proportion.
Example 2: Tara drew a diagram of her bedroom. In the diagram, the longest wall is 8.5 inches, but it actually measures 12.75 feet. What scale factor did Tara use when she made the diagram?
Solution: Convert the units all to inches and then set up a proportion.
Remember: 1 foot = 12 inches
So, 12.75 feet × 12 inches = 153 inches
Scale Factor = x = 153 × 1 ÷ 8.5 = 18
More Scale Factor
Not all scale factors you will be given are in the form 1:x. Often, the 1 will be some other number. When this is the case, use a proportion to solve the problem.
Example 1: Jacob is building a model of a room using a scale factor of 6:200. If the dimensions of the room are 650 cm by 480 cm, what will the dimensions of the model be?
Solution: Set up a proportion and solve. One proportion for each dimension is necessary.
The dimensions of the model are 19.5 cm by 14.4 cm.
Example 2: The scale of a photograph of an organism under a microscope is 75:2. If the photograph has a dimension of 30 mm, how long was the original organism?
Solution: Set up a proportion and solve.
ASK YOUR TEACHER FOR UNIT QUIZ 2.
Working with Similar Figures
In the first part of this unit, you learned about similar figures and how to find their corresponding sides and angles. In this section you will determine if two figures are similar, and what changes you can make to a shape to keep it similar to the original.
Example 1: looking at the two figures below, are they similar? If so, explain how you know. If not, explain what is missing or wrong. The angles marked with the same symbol are equal.
You can see that 3 of the angles in the large figure are equal to their corresponding angles in the smaller figure.
But you cannot state that the other 2 pairs of corresponding angles are equal as there is no evidence to support that. Therefore, you cannot state that the 2 figures are similar.
DRAWING SIMILAR FIGURES
Artists, architects, and planners use scale drawings in their work. The diagrams or models should be in proportion to the actual objects so that others can visualize what the real objects look like accurately.
Example: Use graph paper to draw a figure similar to the one given, with the sides 1.5 times the length of the original. Remember that the corresponding angles must be equal.
The lengths of the sides, starting in the bottom left corner and going clockwise around the figure are: 6 squares, 4 diagonals, 4 squares, 10 diagonals, 18 squares.
The new lengths are: 6 × 1.5 = 9 squares
4 × 1.5 = 6 diagonals
4 × 1.5 = 6 squares
10 × 1.5 = 15 diagonals
18 × 1.5 = 27 squares
Similar triangles are very useful in making calculations and determining measurements. There are certain things to know about triangles before proceeding. Triangles always have 3 sides and three angles. The sum of the angles of a triangle is always 1800.
If two corresponding angles are equal, the third angles will also be equal because the sum must be 1800.
There are several special triangles – an isosceles triangle has 2 sides equal in length, and the two angles opposite these sides are of equal measure. An equilateral triangle has all three sides equal in length and all three angles equal in measure to 600.
Two triangles are similar if any two of the three corresponding angles are congruent, or one pair of corresponding angles is congruent and the corresponding sides beside the angles are proportional. Congruent means the same in size and shape.
Example 2: Kevin notices that a 2 m pole casts a shadow of 5 m, and a second pole casts a shadow of 9.4 m. How tall is the second pole?
Solution: First, always make a diagram if one is not provided. Then confirm that the triangles are similar, and then use a proportion to solve for x.
Notice that 2 of the three corresponding angles are congruent. The third angles are also equal because the angle between the rays of the sun and the poles is the same in both cases. So the triangles are similar.
Now set up a proportion to solve for x.
YOU ARE NOW FINISHED THIS UNIT
ASK YOUR TEACHER FOR THE UNIT #5 TEST