Unit 3: Ratio, proportion and similarity

UNIT 3: PROPORTIONALITY AND SIMILARITY

In this unit participants will look at the concept of

• Ratio and proportion
• Similarity
• Ratios of areas of triangles

LEARNING OUTCOME

• At the end of this Unit, participants should be able to:
• Proof the proportionality theorem
• Apply the proportionality theorem and the converse
• Proof the Similarity theorem
• Apply and use the similarity theorem and the converse

Apply and use midpoint theorem as a special case of proportionality

1. RATIO

Other real-life examples of ratios

Say for instance, you are in a classroom. In the classroom, there are 3 boys and 6 girls.
The ratio of boys to girls is 3/6

It means that there are half as many boys as girls in the classroom. However, the ratio of girls to boys is 6/3

6/3 is equal to 2 and it means that there are two times as many girls as boys in the classroom

DEFINITION OF A RATIO

A RATIO IS A COMPARISON OF TWO QUANTIES OF THE SAME KIND.

The ratio of a to b is   with b ≠ 0
A ratio is an ordered pair of numbers, written a:b, with b ≠ 0

What is a continued ratio? Definition and examples

The ratio of three or more quantities is called continued ratio. For example, the ratio of 4 to 8 to 12 is the continued ratio 4:8:12.

2. PROPORTION

Definition:
Simply put, whenever we put an equal sign between two ratios and the ratio on the left is equal to the ratio on the right, we say that they form a proportion.

Look at the following two ratios. Right now they are just ratios.

Do they form a proportion? They will form a proportion if the ratios are equal, so first put an equal sign between them. Then, you have two choices to check if the ratios are equal
Choice #1: You can just convert both ratios into decimals and see if the decimals are equal.

Since 0.60 ≠ 0.50, the ratios are not equal and therefore do not for a proportion.

Choice #2: You can also tell if two ratios are equal by comparing their cross products. If the cross products are equal, then they form a proportion.

Recall that a cross product is obtained when you multiply the numerator of one fraction by the denominator of another fraction.

3 × 10 = 30 and 5 × 5 = 25

You have worked with triangles before and should already be familiar with the definitions of the different types of these shapes.
Note:

• All triangles have three sides and three angles.
• A triangle can be described by looking at the sizes of its angles, and the lengths of its sides.
• A triangle is named using the letters at the vertices (the order of the letters does not matter).

This is ∆ PQR
The angles can be named as P, Q and R

The sides can be named as PR, PQ and RQ

An altitude (height) of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side.