Testing Similarity with Proportions
Grade 8 math students learn to test polygon similarity using proportions, checking that ratios of ALL corresponding side lengths are equal. If every corresponding side ratio simplifies to the same scale factor k, the figures are similar. Covered in Big Ideas Math, Course 3, Chapter 2: Transformations.
Key Concepts
Property To officially prove two polygons are similar without using transformations, you must test the ratios of ALL their corresponding sides. You create a fraction for each pair of sides: $\frac{\text{Image Side}}{\text{Pre image Side}}$. If every single fraction simplifies to the exact same number (the scale factor $k$), the figures are similar. $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$.
Examples Testing Triangles: Triangle 1 has sides $3, 4, 5$. Triangle 2 has sides $6, 8, 10$. Check ratios: $\frac{6}{3} = 2$, $\frac{8}{4} = 2$, $\frac{10}{5} = 2$. All equal $2$. They are similar! Testing Rectangles: Rectangle A is $4$ by $6$. Rectangle B is $6$ by $8$. Check ratios: $\frac{6}{4} = 1.5$, but $\frac{8}{6} \approx 1.33$. The ratios are NOT equal. They are not similar.
Explanation This is the ultimate test for similarity. A common mistake students make is only checking one pair of sides and assuming the whole shape is similar. You must check every pair! When setting up your fractions, be organized: always put the sides of the second shape on the top (numerator), and the sides of the first shape on the bottom (denominator). Make sure you are matching the shortest side to the shortest side, and the longest to the longest!
Common Questions
How do you test if two polygons are similar using proportions?
Create a fraction for each pair of corresponding sides (image side / pre-image side). If every fraction simplifies to the same number (the scale factor k), the figures are similar. You must check ALL pairs.
What is a scale factor in similarity?
The scale factor is the constant ratio k between corresponding sides of similar figures. For example, if all sides of shape B are twice as long as corresponding sides of shape A, the scale factor is 2.
Why must you check all side ratios to test similarity?
Checking only one pair of sides is not enough. All pairs of corresponding sides must have equal ratios. If even one pair has a different ratio, the figures are not similar.
Which textbook covers testing similarity with proportions for Grade 8?
This topic is in Big Ideas Math, Course 3, Chapter 2: Transformations.
What grade level covers polygon similarity?
Testing polygon similarity with proportions is typically covered in Grade 8 math.