Lesson 12

Practice With Proportional Relationships

  • Let’s find unknown values in proportional relationships.

12.1: Vegetable Garden

These are the plans for a vegetable garden that a school is designing.

Scale: 1 unit = 2.8 ft

Triangle A B C. Length of A B is 4, A C is 3, and B C is 5. At the top of the triangle are 3 potatoes, 6 carrots underneath, then 3 pumpkins at the bottom.

Write at least 3 equivalent ratios or equations using lengths from both the diagram and the full-size garden.

12.2: Card Sort: Corresponding Parts

Your teacher will give you a set of cards. Group them into pairs of similar figures. For each pair, determine:

  1. a similarity statement
  2. the scale factor between the similar figures
  3. the missing lengths

12.3: Quilting Questions

  1. Here is a quilt design made of right isosceles triangles. The smallest squares in the center have an area of 1 square unit. Find the dimensions of the triangles.
    A small square with isosceles right triangles on its sides, forming a larger square. Isosceles right triangles are on those sides, forming another square. There is one more iteration of the pattern.
  2. Are the triangles similar? If so, what are the scale factors?
  3. This quilt is meant for a baby (1 unit = 6 inches). To make a quilt for a queen-size bed, it needs to be 90 inches wide. What dimensions should the center squares of the big quilt have to reach that width?

  1. Here is a quilt design made of right triangles. Find as many different size triangles as you can.
    A shape made up of many right triangles. Diamond-shape C F Q L is divided into 16 equal-sized right triangles. Right triangles C L D and C F E extend out from the diamond-shape like wings.
  2. Write similarity statements for 2 pairs of triangles.
  3. The smallest triangles have legs 2 units and 3 units long. Write some equivalent ratios or equations that will help you determine the dimensions of triangle \(LCQ\). Use your equivalent ratios or equations to find the dimensions of triangle \(LCQ\).


When 2 figures are similar, there are lots of equivalent ratios between the triangles and within the triangles. We can use those relationships to find missing lengths. For example, if we know that triangle \(ABC\) is similar to triangle \(DEF\), we know that pairs of corresponding side lengths are in the same proportion. \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\)

Two similar triangles A B C and D E F. Length of A B is 3, B C is 4, A C is 5, and E F is 6.

We also know that pairs of side lengths in one triangle are in the same proportion as pairs of side lengths in the other triangle.




We can use these equivalent ratios to find unknown side lengths. Which equivalent ratios would work to find \(DE\) and \(DF\)?

We can use \(\frac{AB}{BC}=\frac{DE}{EF}\) to find \(DE\). Then \(\frac34=\frac{DE}{6}\), which gives \(DE=4.5\).

Or we can use \(\frac{AB}{DE}=\frac{BC}{EF}\) to find \(DE\). In this case, we get \(\frac{3}{DE}=\frac46\), which also gives \(DE=4.5\).

Glossary Entries

  • similar

    One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

    Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).

    Triangle ABC, rotated and then dilated.