Introduction

Similar Figures and Proportions is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with similar figures and proportions.

What Is Similar Figures and Proportions?

Similar Figures and Proportions means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Similar Figures and Proportions

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Triangles 1 and 2 are similar right triangles. What is the length of the missing side in Triangle 2?

Visual Model 1

  • A. \(7\) cm
  • B. \(8\) cm
  • C. \(9\) cm
  • D. \(9.5\) cm

Why it works: The scale factor from Triangle 1 to Triangle 2 is \(\frac{12}{8}=1.5\). The missing side is \(6 \times 1.5 = 9\) cm.

Answer: \(9\) cm

Visual Model 2

Question: The two rectangles are similar. What is the scale factor from the original to the scaled rectangle?

Visual Model 2

  • A. \(0.71\)
  • B. \(1.4\)
  • C. \(1.5\)
  • D. \(2.8\)

Why it works: The scale factor is \(\frac{28}{20}=1.4\). We can verify: \(10 \times 1.4 = 14\) \checkmark

Answer: \(1.4\)

Worked Examples

Example 1

Question: Two similar right triangles are shown. The left triangle has legs \(5\) ft (horizontal) and \(4\) ft (vertical). The right triangle has a vertical leg of \(3.2\) ft. Find the horizontal leg of the right triangle.

Example 1

  • A. \(2.56\) ft
  • B. \(3.2\) ft
  • C. \(4\) ft
  • D. \(5\) ft
  1. The vertical leg of the left triangle is \(4\) ft, and the right triangle's vertical leg is \(3.2\) ft.
  2. Scale factor \(= \frac{3.2}{4}=0.8\).
  3. The horizontal leg of the right triangle is \(5 \times 0.8 = 4\) ft.

Answer: \(4\) ft

Example 2

Question: The figure on the left is scaled by a factor of \(1.5\) to create the figure on the right. If the height of the left figure is \(2\) cm, what is the height of the scaled figure?

Example 2

  • A. \(2\) cm
  • B. \(2.5\) cm
  • C. \(3\) cm
  • D. \(4\) cm
  1. All linear dimensions scale by the scale factor.
  2. Height \(= 2 \times 1.5 = 3\) cm.

Answer: \(3\) cm

Example 3

Question: Two circles are similar (scaled). Circle 1 has radius \(4\) cm. Circle 2 has radius \(6\) cm. What is the ratio of their areas?

Example 3

  • A. \(4:6\)
  • B. \(16:36\)
  • C. \(2:3\)
  • D. \(8:27\)
  1. The linear scale factor is \(\frac{4}{6}=\frac{2}{3}\).
  2. Areas scale by the square: \((\frac{2}{3})^2=\frac{4}{9}\), so the ratio is \(\frac{\pi(4)^2}{\pi(6)^2}=\frac{16\pi}{36\pi}=\frac{16}{36}\), or \(16:36\).

Answer: \(16:36\)

Real-World Word Problems

Problem 1

Question: Two similar rectangles have a scale factor of \(2:5\). If the smaller rectangle has a perimeter of \(28\) inches, what is the perimeter of the larger rectangle?

  • A. \(35\) inches
  • B. \(56\) inches
  • C. \(70\) inches
  • D. \(140\) inches

Why it works: The perimeter scales with the same factor as the sides. If the scale factor is \(2:5\), then \(\frac{\text{smaller}}{\text{larger}}=\frac{2}{5}\), so \(\text{larger}=28\times\frac{5}{2}=70\) inches.

Answer: \(70\) inches

Problem 2

Question: In a map, a scale of \(1\) inch \(= 50\) miles is used. On the map, two cities are \(3.5\) inches apart. What is the actual distance between the cities?

  • A. \(150\) miles
  • B. \(175\) miles
  • C. \(200\) miles
  • D. \(350\) miles

Why it works: Using the scale: \(3.5 \text{ inches} \times 50 \text{ miles/inch} = 175\) miles.

Answer: \(175\) miles

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Two triangles are similar. The sides of the smaller triangle are \(3\), \(4\), and \(5\). The longest side of the larger triangle is \(15\). What is the length of the shortest side of the larger triangle?

  • A. \(6\)
  • B. \(8\)
  • C. \(9\)
  • D. \(12\)

Question 2

Two similar triangles have corresponding sides in the ratio \(3:7\). If an angle in the smaller triangle measures 45°, what is the measure of the corresponding angle in the larger triangle?

  • A. 45°
  • B. 52°
  • C. 105°
  • D. 315°

Question 3

Two similar isosceles triangles have areas in the ratio \(4:25\). What is the ratio of their corresponding sides?

  • A. \(2:5\)
  • B. \(4:25\)
  • C. \(2:25\)
  • D. \(16:625\)

Question 4

A photograph has dimensions \(6\) inches by \(8\) inches. It is enlarged so that the longer side becomes \(20\) inches. What is the length of the shorter side after enlargement?

  • A. \(12\) inches
  • B. \(14\) inches
  • C. \(15\) inches
  • D. \(16\) inches

Question 5

If \(\frac{x}{6}=\frac{12}{8}\), what is the value of \(x\)?

  • A. \(4\)
  • B. \(8\)
  • C. \(9\)
  • D. \(16\)

Question 6

Two similar figures have a scale factor of \(3:4\). If the smaller figure has a side length of \(9\) cm, what is the length of the corresponding side in the larger figure?

  • A. \(6.75\) cm
  • B. \(12\) cm
  • C. \(13.5\) cm
  • D. \(27\) cm
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(9\)

The scale factor is \(15\div5=3\). The shortest side of the larger triangle is \(3\times3=9\).

Question 2

Answer: 45°

Corresponding angles in similar figures are congruent, regardless of the scale factor. The corresponding angle is also 45°.

Question 3

Answer: \(2:5\)

Area scales by the square of the linear scale factor. If areas are \(4:25\), then the linear scale factor is \(\sqrt{4}:\sqrt{25}=2:5\).

Question 4

Answer: \(15\) inches

The scale factor is \(\frac{20}{8}=2.5\). The shorter side is \(6 \times 2.5 = 15\) inches.

Question 5

Answer: \(9\)

Cross-multiply: \(8x = 12 \times 6 = 72\), so \(x = \frac{72}{8} = 9\).

Question 6

Answer: \(12\) cm

If the scale factor is \(3:4\), then \(\frac{\text{smaller}}{\text{larger}}=\frac{3}{4}\). So \(\frac{9}{\text{larger}}=\frac{3}{4}\), giving \(\text{larger}=9\times\frac{4}{3}=12\) cm.

Timed practice quizzes

Related Quizzes

After studying Similar Figures and Proportions, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Similar Figures and Proportions becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.