Introduction

Similarity and Dilations is an important Grade 8 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 similarity and dilations.

What Is Similarity and Dilations?

Similarity and Dilations 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 Similarity and Dilations

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: Two triangles are shown below. Triangle \(ABC\) has sides \(3, 4, 5\). Triangle \(DEF\) has sides \(6, 8, 10\). Are these triangles similar?

Visual Model 1

  • A. No, because the sides are different lengths
  • B. Yes, each side of \(DEF\) is twice the corresponding side of \(ABC\)
  • C. No, the angles are not equal
  • D. Cannot be determined

Why it works: Compute ratios of corresponding sides: \(\frac{DE}{AB} = \frac{6}{3} = 2\), \(\frac{DF}{AC} = \frac{8}{4} = 2\), \(\frac{EF}{BC} = \frac{10}{5} = 2\). All ratios are constant (= 2), confirming similarity by SSS Similarity. Angles are also congruent in similar figures.

Answer: Yes, each side of \(DEF\) is twice the corresponding side of \(ABC\)

Visual Model 2

Question: Two similar triangles are shown. Triangle \(ABC\) has sides \(4 \text{ cm}\) and \(5 \text{ cm}\). Triangle \(DEF\) has a corresponding side of \(8 \text{ cm}\). Find the length of the side in \(DEF\) corresponding to the \(5 \text{ cm}\) side in \(ABC\).

Visual Model 2

  • A. \(10 \text{ cm}\)
  • B. \(6.5 \text{ cm}\)
  • C. \(9 \text{ cm}\)
  • D. \(2.5 \text{ cm}\)

Why it works: Scale factor from \(ABC\) to \(DEF\) is \(\frac{8}{4} = 2\). The corresponding side is \(5 \times 2 = 10 \text{ cm}\).

Answer: \(10 \text{ cm}\)

Worked Examples

Example 1

Question: Figure \(P\) has area \(16 \text{ m}^2\) and Figure \(Q\) (similar to \(P\)) has area \(64 \text{ m}^2\). What is the scale factor from \(P\) to \(Q\)?

Example 1

  • A. \(2\)
  • B. \(4\)
  • C. \(8\)
  • D. \(16\)
  1. If area ratio is \(\frac{64}{16} = 4\), then the linear scale factor is \(\sqrt{4} = 2\).

Answer: \(2\)

Example 2

Question: Two triangles both have angles measuring 45°, 60°, and 75°. Are these triangles necessarily similar?

Example 2

  • A. No, the sides must also be proportional
  • B. No, only two angles are given
  • C. Yes, angles alone determine similarity (AAA)
  • D. Only if the triangles are congruent
  1. If two triangles have all three corresponding angles congruent, they are similar by AAA (Angle-Angle-Angle).
  2. Here, both triangles have the same three angles, so they must be similar regardless of side lengths.

Answer: Yes, angles alone determine similarity (AAA)

Example 3

Question: Point \(A\) is at \((2, 3)\) and is dilated by a scale factor of \(2\) from the origin. Where is the dilated point \(A'\)?

Example 3

  • A. \((1, 1.5)\)
  • B. \((4, 6)\)
  • C. \((2, 3)\)
  • D. \((2, 5)\)
  1. Multiply each coordinate by the scale factor: \((2 \times 2, 3 \times 2) = (4, 6)\).

Answer: \((4, 6)\)

Real-World Word Problems

Problem 1

Question: A triangle has a perimeter of \(15 \text{ inches}\). If the triangle is dilated by a scale factor of \(\frac{1}{2}\), what is the perimeter of the dilated triangle?

  • A. \(30 \text{ inches}\)
  • B. \(7.5 \text{ inches}\)
  • C. \(15 \text{ inches}\)
  • D. \(10 \text{ inches}\)

Why it works: Perimeter scales linearly with the scale factor. \(15 \times \frac{1}{2} = 7.5\) inches. A scale factor less than \(1\) creates a reduction.

Answer: \(7.5 \text{ inches}\)

Problem 2

Question: A photograph is \(10 \text{ inches}\) wide. A thumbnail is \(0.5 \text{ inches}\) wide. What is the scale factor from the photograph to the thumbnail?

  • A. \(20\)
  • B. \(0.05\)
  • C. \(0.5\)
  • D. \(2\)

Why it works: Scale factor is \(\frac{0.5}{10} = 0.05\), a reduction.

Answer: \(0.05\)

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

A figure is dilated by a scale factor of \(3\). How does the length of each side change?

  • A. Divided by \(3\)
  • B. Multiplied by \(3\)
  • C. Unchanged
  • D. Increased by \(3\) inches

Question 2

Rectangle \(PQRS\) has length \(10 \text{ m}\) and width \(6 \text{ m}\). Rectangle \(TUVW\) has length \(5 \text{ m}\) and width \(3 \text{ m}\). What is the scale factor from \(PQRS\) to \(TUVW\)?

  • A. \(2\)
  • B. \(\frac{1}{2}\)
  • C. \(3\)
  • D. \(\frac{1}{3}\)

Question 3

Triangle \(X\) is similar to Triangle \(Y\). In Triangle \(X\), one angle measures 60°. What is the measure of the corresponding angle in Triangle \(Y\)?

  • A. 30°
  • B. 60°
  • C. 120°
  • D. Cannot be determined

Question 4

A square has area \(9 \text{ cm}^2\). If the square is dilated by a scale factor of \(3\), what is the area of the dilated square?

  • A. \(27 \text{ cm}^2\)
  • B. \(81 \text{ cm}^2\)
  • C. \(9 \text{ cm}^2\)
  • D. \(36 \text{ cm}^2\)

Question 5

Triangles \(MNO\) and \(PQR\) are similar. If \(MN = 6\), \(NO = 8\), and \(PQ = 9\), find \(QR\).

  • A. \(10\)
  • B. \(12\)
  • C. \(6.75\)
  • D. \(9.5\)

Question 6

A segment has length \(20 \text{ cm}\) and is dilated by a scale factor of \(0.4\). What is the new length?

  • A. \(8 \text{ cm}\)
  • B. \(20 \text{ cm}\)
  • C. \(24 \text{ cm}\)
  • D. \(50 \text{ cm}\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: Multiplied by \(3\)

Dilation with a scale factor of \(k\) multiplies every distance (and side length) by \(k\). Here each side becomes \(3\) times longer. Distractor D confuses scale factor with absolute addition.

Question 2

Answer: \(\frac{1}{2}\)

Scale factor is the ratio of a side in \(TUVW\) to the corresponding side in \(PQRS\): \(\frac{5}{10} = \frac{1}{2}\).

Question 3

Answer: 60°

In similar figures, corresponding angles are congruent. The angle measure does not change with dilation.

Question 4

Answer: \(81 \text{ cm}^2\)

Area scales by the square of the scale factor: original side is \(3 \text{ cm}\), new side is \(9 \text{ cm}\), new area is \(81 \text{ cm}^2\) (or \(9 \times 3^2 = 81\)).

Question 5

Answer: \(12\)

Scale factor is \(\frac{PQ}{MN} = \frac{9}{6} = 1.5\). So \(QR = NO \times 1.5 = 8 \times 1.5 = 12\).

Question 6

Answer: \(8 \text{ cm}\)

\(20 \times 0.4 = 8 \text{ cm}\). A scale factor between 0 and 1 creates a reduction.

Timed practice quizzes

Related Quizzes

After studying Similarity and Dilations, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 8 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

Similarity and Dilations 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.