Similarity Concepts

Grade 8 · geometry · 103 practice problems · read aloud

🔊 Listen to this explanation

Similarity Concepts in Geometry

🔍 What is Similarity?

Two figures are similar if they have the same shape but not necessarily the same size. Think of a photo and its enlargement! They are useful for creating scale models, maps, and solving real-world problems involving indirect measurement.

For two shapes to be similar:

  • Corresponding angles are congruent (equal).
  • Corresponding sides are proportional (in the same ratio).

The symbol for similarity is ∼. If triangle ABC is similar to triangle DEF, we write ΔABC ∼ ΔDEF.

📝 Step-by-Step Guide to Solving Similarity Problems

  1. Identify corresponding angles and sides. Matching order in the similarity statement (e.g., ΔABC ∼ ΔDEF) helps!
  2. Set up a proportion using the ratios of corresponding sides.
  3. Solve for the unknown side length using cross-multiplication.
  4. Check that your answer makes sense (e.g., a side shouldn't be longer than the sum of the other two).

📐 Visual Examples

Example 1: Finding a Missing Side

Given ΔJKL ∼ ΔPQR, KL corresponds to QR. If KL = 6, QR = 9, and JL = 10, find PR.

  1. Set up the proportion with corresponding sides: KL/QR = JL/PR
  2. Plug in the known values: 6/9 = 10/PR
  3. Cross-multiply: 6 * PR = 9 * 10 → 6 * PR = 90
  4. Solve for PR: PR = 90 / 6 = 15

Example 2: Similar Triangles in a Diagram

A tree casts a 20 ft shadow. A 5 ft tall person nearby casts a 4 ft shadow. The triangles formed by the objects and their shadows are similar. How tall is the tree?

  1. Set up the proportion: (Tree Height)/(Person's Height) = (Tree's Shadow)/(Person's Shadow)
  2. This gives us: T / 5 = 20 / 4
  3. Simplify: T / 5 = 5
  4. Solve: T = 5 * 5 = 25 feet

⚠️ Common Mistakes

  • Mismatching Sides: The most common error! Always ensure sides correspond. Use the order of letters in the similarity statement as your guide.
  • Incorrect Proportions: Writing the proportion as a/b = c/d instead of a/c = b/d. Keep the figures separate on each side of the equal sign.
  • Assuming Similarity: Not all shapes that look the same are similar! You must confirm that both angle and side conditions are met (or use a similarity theorem like AA).

💡 Tips & Tricks

  • AA Theorem is Your Friend: For triangles, if just two pairs of corresponding angles are congruent, the triangles are similar! You don't always need all the sides.
  • Memory Aid: "CPStp" - Corresponding Parts of Similar triangles are proportional.
  • Scale Factor: Find the ratio between one pair of corresponding sides first. This "scale factor" can then be multiplied by any side in the smaller figure to find its corresponding side in the larger figure.

🎯 Practice Suggestions

  • Start by identifying similar figures in your textbook or online, justifying why they are similar (e.g., AA, SSS, SAS similarity).
  • Solve at least 5 problems finding missing side lengths using proportions.
  • Create your own word problems using real-life objects (like the tree example) to make the concept stick.
  • Use online tools or geometry software to drag and resize similar shapes and see the relationships in real-time.

Practice problems

6 of the 103, worked through step by step — try them before opening the answer.

1 √(8² + 6²) = ?

Hint: This involves finding the square root of a sum of squares. Consider what mathematical theorem applies to this pattern.

Show the answer

Answer: 10

  1. Calculate 8 squared: 8² = 64
  2. Calculate 6 squared: 6² = 36
  3. Add the squares: 64 + 36 = 100
  4. Take the square root: √100 = 10

The answer is 10.

2 √(15² - 9²) = ?

Hint: This involves applying the Pythagorean theorem to find a missing side length in a right triangle

Show the answer

Answer: 12

  1. Calculate the squares: 15² = 225 and 9² = 81
  2. Subtract the squares: 225 - 81 = 144
  3. Take the square root: √144 = 12

The answer is 12.

3 (3 × 10⁸) ÷ (6 × 10⁴) = ?

Hint: When dividing numbers in scientific notation, handle the coefficients and powers of ten separately. Remember that dividing powers with the same base means subtracting their exponents.

Show the answer

Answer: 5000

  1. Break the expression into two parts — the numerical coefficients and the powers of ten. (3 × 10⁸) ÷ (6 × 10⁴) = (3 ÷ 6) × (10⁸ ÷ 10⁴)
  2. Simplify the numerical part. 3 ÷ 6 = 3/6 = 1/2 = 0.5
  3. Simplify the powers of ten using the rule: 10⁸ ÷ 10⁴ = 10^(8 - 4) = 10⁴. So now we have: 0.5 × 10⁴
  4. Multiply 0.5 by 10⁴. 0.5 × 10⁴ = 0.5 × 10000 = 5000
  5. Final answer. 5000 That’s the result.

Let's solve step by step. We have: (3 × 10⁸) ÷ (6 × 10⁴)

4 (4.5 × 10⁶) ÷ (1.5 × 10²) = ?

Hint: Separate the decimal numbers and the powers of ten, then simplify each part before combining them.

Show the answer

Answer: 30000

  1. Write the division of the two numbers in scientific notation: (4.5 × 10⁶) ÷ (1.5 × 10²)
  2. Separate the problem into two parts: (4.5 ÷ 1.5) and (10⁶ ÷ 10²)
  3. Calculate 4.5 ÷ 1.5 = 3
  4. Calculate 10⁶ ÷ 10² = 10^(6-2) = 10⁴
  5. Combine the results: 3 × 10⁴
  6. Convert 3 × 10⁴ to standard form: 3 × 10000 = 30000

The answer is 30000.

5 (4.2 × 10^5) ÷ (7 × 10^2) = ?

Hint: Consider how to handle the division of numbers in scientific notation separately from the powers of ten.

Show the answer

Answer: 600

  1. Separate the problem into two parts: (4.2 ÷ 7) and (10^5 ÷ 10^2).
  2. Calculate 4.2 ÷ 7 = 0.6.
  3. Calculate 10^5 ÷ 10^2 = 10^(5-2) = 10^3.
  4. Multiply the results: 0.6 × 10^3 = 0.6 × 1000 = 600.

The answer is 600.

6 (2.4 × 10³) × (3.5 × 10⁻²) = ?

Hint: When multiplying numbers in scientific notation, multiply the coefficients and add the exponents separately.

Show the answer

Answer: 84

  1. Multiply the coefficients: 2.4 × 3.5 = 8.4
  2. Add the exponents: 3 + (-2) = 1
  3. Combine the results: 8.4 × 10¹
  4. Convert to standard form: 8.4 × 10 = 84

The answer is 84.

Practise this topic — 10 free problems, no signup →