Similar Figures

Grade 7 · geometry · 109 practice problems · read aloud

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What are Similar Figures? 🔍

Similar figures are shapes that have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. This concept is super useful for solving real-world problems like making scale models, reading maps, and finding heights of tall objects without climbing them!

How to Work with Similar Figures

  1. Identify corresponding angles and sides. Matching order is key!
  2. Check that all corresponding angles are equal.
  3. Set up a proportion using the corresponding sides.
  4. Cross-multiply and solve for the missing side length.

Worked Examples

Example 1: Finding a Missing Side

Triangles ABC and DEF are similar. Side AB = 4 cm, BC = 6 cm, and the corresponding side DE = 10 cm. Find EF.

Step 1: Set up the proportion: AB/DE = BC/EF

Step 2: Plug in known values: 4/10 = 6/EF

Step 3: Cross-multiply: 4 × EF = 10 × 6 → 4EF = 60

Step 4: Solve: EF = 60 ÷ 4 = 15 cm

Example 2: Scale Factor

Two similar rectangles. The first is 3 m by 5 m, the second is 6 m by x m.

Step 1: Find the scale factor using one pair of sides: 6 ÷ 3 = 2

Step 2: Apply the scale factor to find x: 5 × 2 = 10 m

So the missing side is 10 meters.

Common Mistakes to Avoid

Mixing up corresponding sides: Always match the sides in the same order. Label your shapes clearly!

Angle mix-up: Remember, the angles must be equal for figures to be similar. If angles aren't equal, the shapes aren't similar!

Setting up proportions wrong: Make sure you're comparing corresponding parts. Don't mix and match willy-nilly!

Tips & Tricks

Memory Aid: "Same Shape, Sides Scale" – the three S's of similarity!

Shortcut: Find the scale factor first, then multiply all sides by it.

Strategy: When writing proportions, keep corresponding parts in the same position (e.g., top/top = bottom/bottom).

How to Practice

  • Draw your own similar figures and find missing lengths.
  • Look for similar shapes in your environment (windows, books, tiles).
  • Practice with online quizzes that give instant feedback.
  • Work with a partner to create problems for each other.

Practice problems

6 of the 109, worked through step by step — try them before opening the answer.

1 A map has a scale of 1:25000. If two landmarks are 8 cm apart on the map, what is their actual distance in kilometers?

Hint: Remember that scale ratios relate map distance to actual distance. Convert units carefully when moving from centimeters to kilometers.

Show the answer

Answer: 2

  1. Understand the scale. A scale of 1:25000 means that 1 cm on the map represents 25000 cm in real life.
  2. Find the actual distance in centimeters. The map distance is 8 cm. Actual distance = map distance × scale factor Actual distance = 8 cm × 25000 Actual distance = 200000 cm.
  3. Convert centimeters to meters. We know 1 m = 100 cm, so: 200000 cm ÷ 100 = 2000 m.
  4. Convert meters to kilometers. We know 1 km = 1000 m, so: 2000 m ÷ 1000 = 2 km. Final answer: The actual distance is 2 kilometers.

2 A map has a scale of 1:25000. If two towns are 8 cm apart on the map, what is the actual distance between them in kilometers?

Hint: Remember that scale ratios relate map distance to actual distance. Convert units carefully when moving from centimeters to kilometers.

Show the answer

Answer: 2

  1. Understand the scale The map scale is 1:25000. This means that 1 unit on the map represents 25000 of the same units in real life.
  2. Apply the scale to the map distance The map distance between the two towns is 8 cm. So, the actual distance in centimeters is: 8 cm × 25000 = 200000 cm.
  3. Convert centimeters to meters We know 1 meter = 100 cm. So, actual distance in meters = 200000 cm ÷ 100 = 2000 meters.
  4. Convert meters to kilometers We know 1 kilometer = 1000 meters. So, actual distance in kilometers = 2000 m ÷ 1000 = 2 km. Final answer: The actual distance between the two towns is 2 kilometers.

3 A map has a scale of 1:25000. If two cities are 8 cm apart on the map, what is the actual distance between them in kilometers?

Hint: Consider how the scale ratio relates map distance to actual distance. Remember to convert units appropriately at the end.

Show the answer

Answer: 2

  1. Understand the scale. A scale of 1:25000 means that 1 cm on the map represents 25000 cm in real life.
  2. Find the actual distance in centimeters. On the map, the cities are 8 cm apart. Actual distance in cm = map distance × scale factor Actual distance in cm = 8 × 25000 = 200000 cm.
  3. Convert centimeters to meters. We know 1 m = 100 cm, so: Actual distance in m = 200000 / 100 = 2000 m.
  4. Convert meters to kilometers. We know 1 km = 1000 m, so: Actual distance in km = 2000 / 1000 = 2 km. Final answer: The actual distance between the cities is 2 kilometers.

4 A rectangular park measures 120 m by 80 m. If a scale drawing uses 1 cm to represent 20 m, what are the dimensions of the drawing in cm?

Hint: To find scale drawing dimensions, divide each actual measurement by the scale factor. For example, if a 100 m length is represented at 1 cm : 25 m scale, the drawing length would be 100 ÷ 25 = 4 cm.

Show the answer

Answer: 6 cm by 4 cm

  1. Understand the scale. The scale is 1 cm represents 20 m. This means: 1 cm on the drawing = 20 m in real life.
  2. Find the length in cm. Real length = 120 m. We know 20 m = 1 cm, so: 120 m / 20 m = 6 So length on drawing = 6 cm.
  3. Find the width in cm. Real width = 80 m. 80 m / 20 m = 4 So width on drawing = 4 cm.
  4. Final answer. Dimensions of the drawing: 6 cm by 4 cm. This makes sense because dividing each real dimension by 20 m (the scale factor) gives the drawing dimension in cm.

Let's solve this step by step.

5 A rectangular garden measures 18 m by 12 m. A scale drawing uses a scale of 1:150. What is the area of the garden in the scale drawing in cm²?

Hint: Convert the real dimensions to the scale dimensions using the given ratio, then calculate the area of the scaled rectangle.

Show the answer

Answer: 96

  1. Understand the scale** The scale is 1:150. This means 1 unit on the drawing represents 150 units in reality. Both length and width will be scaled down by the factor 150. --- **
  2. Convert real garden dimensions to cm** Real length = 18 m Real width = 12 m 1 m = 100 cm, so: Length in cm = 18 × 100 = 1800 cm Width in cm = 12 × 100 = 1200 cm --- **
  3. Apply the scale to find drawing dimensions in cm** Scale factor (from real to drawing) = 1/150. Drawing length = 1800 cm × (1/150) = 1800 / 150 = 12 cm Drawing width = 1200 cm × (1/150) = 1200 / 150 = 8 cm --- **
  4. Find the area in the drawing** Area in drawing = length in drawing × width in drawing = 12 cm × 8 cm = 96 cm² --- **
  5. Conclusion** The area of the garden in the scale drawing is 96 cm². --- **Final answer:** 96

Let's go step-by-step. --- **

6 A scale drawing of a rectangular room has dimensions 6 cm × 9 cm. If the scale is 1:150, what is the actual area of the room in square meters?

Hint: First find the actual dimensions by applying the scale factor, then calculate the area using those dimensions.

Show the answer

Answer: 121.5

  1. Find the actual length using the scale factor 1:150 Actual length = 9 cm × 150 = 1350 cm
  2. Find the actual width using the scale factor 1:150 Actual width = 6 cm × 150 = 900 cm
  3. Convert centimeters to meters Actual length = 1350 cm ÷ 100 = 13.5 m Actual width = 900 cm ÷ 100 = 9 m
  4. Calculate the actual area Area = length × width = 13.5 m × 9 m = 121.5 m²

The answer is 121.5.

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