Solved Exercises on Understanding Similar Figures in Grade 8

Master similar figures: ratios, proportions, scale factors, and geometric relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Similar Figures
Exercise 1
Determine if rectangles ABCD and EFGH are similar. Rectangle ABCD has dimensions 6 cm × 4 cm, and rectangle EFGH has dimensions 9 cm × 6 cm.
Definition:

Similar Figures: Figures with the same shape but not necessarily the same size

Corresponding Angles: Angles that occupy the same relative position in similar figures are equal

Corresponding Sides: Sides that are in the same relative position in similar figures are proportional

Similarity Check Method:
  1. Identify corresponding sides in both figures
  2. Write ratios of corresponding sides
  3. Check if all ratios are equal
  4. Verify that corresponding angles are equal
  5. Conclude if figures are similar
Rectangle ABCD
6 cm × 4 cm
Rectangle EFGH
9 cm × 6 cm
Ratio
3:2
Step 1: Identify Corresponding Sides

Long side of ABCD corresponds to long side of EFGH: 6 cm and 9 cm

Short side of ABCD corresponds to short side of EFGH: 4 cm and 6 cm

Step 2: Write Ratios of Corresponding Sides

Ratio of long sides: 6/9 = 2/3

Ratio of short sides: 4/6 = 2/3

Step 3: Check Proportionality

Since both ratios equal 2/3, the sides are proportional

Step 4: Verify Angles

All angles in rectangles are 90°, so corresponding angles are equal

Step 5: Conclusion

Since corresponding sides are proportional and corresponding angles are equal, the rectangles are similar

Rectangles are similar (ratio 2:3)
Final answer:

Yes, rectangles ABCD and EFGH are similar with a scale factor of 2:3

Applied rules:

• Similarity Criteria: Corresponding sides proportional AND corresponding angles equal

• Proportion Check: All ratios of corresponding sides must be equal

• Scale Factor: The constant ratio of corresponding sides

2 Finding Missing Side Lengths
Exercise 2
Triangles ABC and DEF are similar. In triangle ABC, AB = 8 cm, BC = 6 cm, and AC = 10 cm. In triangle DEF, DE = 4 cm. Find the lengths of EF and DF.
Definition:

Proportional Sides: In similar figures, the ratio of corresponding sides is constant

Scale Factor: The ratio of any pair of corresponding sides

Corresponding Parts: Parts that have the same relative position in similar figures

Triangle ABC
AB=8, BC=6, AC=10
Triangle DEF
DE=4, EF=?, DF=?
Scale Factor
1:2
Step 1: Find the Scale Factor

Since triangles are similar, corresponding sides are proportional

AB corresponds to DE: 8/4 = 2

Scale factor from ABC to DEF is 1:2 (DEF is half the size of ABC)

Step 2: Find EF

BC corresponds to EF

EF = BC × (scale factor) = 6 × (1/2) = 3 cm

Step 3: Find DF

AC corresponds to DF

DF = AC × (scale factor) = 10 × (1/2) = 5 cm

Step 4: Verify Proportionality

AB/DE = 8/4 = 2, BC/EF = 6/3 = 2, AC/DF = 10/5 = 2

All ratios equal 2, confirming the triangles are similar

EF = 3 cm, DF = 5 cm
Final answer:

The lengths of EF and DF are 3 cm and 5 cm respectively

Applied rules:

• Proportionality: Corresponding sides in similar figures are proportional

• Scale Factor: Use one known pair to find the scale factor

• Consistency: All corresponding sides must have the same ratio

3 Scale Factor Applications
Exercise 3
A map has a scale of 1:50,000. If the distance between two cities on the map is 4.5 cm, what is the actual distance between the cities?
Definition:

Scale Factor: The ratio of the size of a drawing/model to the actual size

Map Scale: A ratio that shows how distances on the map relate to actual distances

Proportional Reasoning: Using ratios to solve real-world problems

Map Scale
1:50,000
Map Distance
4.5 cm
Actual Distance
225,000 cm = 2.25 km
Step 1: Understand the Scale

Scale 1:50,000 means 1 cm on the map represents 50,000 cm in reality

Step 2: Set Up Proportion

Map distance / Actual distance = Scale factor

4.5 cm / Actual distance = 1 / 50,000

Step 3: Solve for Actual Distance

Actual distance = 4.5 × 50,000 = 225,000 cm

Step 4: Convert Units

225,000 cm = 2,250 m = 2.25 km

Step 5: Verify Reasonableness

4.5 cm on map representing 2.25 km is reasonable for a city-to-city distance

Actual distance = 2.25 km
Final answer:

The actual distance between the cities is 2.25 km

Applied rules:

• Scale Factor: Map distance / Actual distance = Scale ratio

• Unit Conversion: Convert to appropriate units for the answer

• Proportional Reasoning: Set up and solve proportions

Rules and methods, laws,...
\(\frac{\text{side}_1}{\text{side}_2} = \frac{\text{side}_3}{\text{side}_4}\)
Proportion for Similar Figures
Scale Factor
\(k = \frac{\text{new}}{\text{original}}\)
Size ratio
Perimeter Ratio
\(k = \frac{P_1}{P_2}\)
Same as scale factor
Area Ratio
\(k^2 = \frac{A_1}{A_2}\)
Square of scale factor
Key definitions:

Similar Polygons: Polygons with corresponding angles equal and corresponding sides proportional

Scale Factor: The constant of proportionality between corresponding sides

Corresponding Parts: Parts that have the same relative position in similar figures

Indirect Measurement: Using similar figures to find distances that are difficult to measure directly

Complete methodology:
  1. Identify Similar Figures: Confirm that figures have the same shape
  2. Match Corresponding Parts: Identify corresponding sides and angles
  3. Set Up Proportions: Write ratios of corresponding sides
  4. Solve Proportions: Use cross multiplication to find unknown values
  5. Verify Solution: Check that all ratios are consistent
  6. Apply Context: Interpret results in real-world context if applicable
Tip 1: Always write ratios in the same order: (figure 1 side)/(figure 2 side).
Tip 2: In similar figures, corresponding angles are always equal.
Tip 3: The ratio of perimeters equals the scale factor.
Tip 4: The ratio of areas equals the square of the scale factor.
Common errors: Mixing up corresponding sides, forgetting to square for area ratios, inconsistent units.
Exam preparation: Practice with various polygon types, understand the relationship between scale factors and area/perimeter ratios.
Formulas to know by heart:

• Similar figures: Corresponding angles equal, corresponding sides proportional

• Scale factor: k = new measurement / original measurement

• Perimeter ratio: Same as scale factor

• Area ratio: Square of scale factor

• Volume ratio: Cube of scale factor

Exercise with Visualization: Similarity Properties
Exercise 6: Similarity Ratios Analysis
Compare the properties of similar figures:
Original triangle: sides 3, 4, 5; Perimeter = 12; Area = 6
Similar triangle: sides 6, 8, 10; Perimeter = 24; Area = 24
Show how ratios are related.

Analysis: The chart shows how similarity affects different properties.

  • Side ratio: 3:6 = 1:2
  • Perimeter ratio: 12:24 = 1:2 (same as scale factor)
  • Area ratio: 6:24 = 1:4 (square of scale factor)
  • All corresponding angles are equal!

Questions & Answers

Question: How can I tell if two figures are similar? What do I need to check?

Answer: To determine if two figures are similar, you need to check two conditions:

  1. Corresponding Angles Equal: All pairs of corresponding angles must have equal measures
  2. Corresponding Sides Proportional: The ratios of all pairs of corresponding sides must be equal

For triangles specifically, you can use shortcuts:

  • AA (Angle-Angle): If two angles of one triangle equal two angles of another, the triangles are similar
  • SSS (Side-Side-Side): If all three pairs of corresponding sides are proportional, the triangles are similar
  • SAS (Side-Angle-Side): If two pairs of corresponding sides are proportional and the included angles are equal, the triangles are similar

For polygons with more than three sides, you must check both conditions: equal corresponding angles AND proportional corresponding sides.

If both conditions are met, the figures are similar!

Question: What's the difference between the ratio of sides, perimeters, and areas for similar figures?

Answer: These ratios are related but different:

  • Ratio of Corresponding Sides: This is the scale factor (k)
  • Ratio of Perimeters: This equals the scale factor (k)
  • Ratio of Areas: This equals the square of the scale factor (k²)
  • Ratio of Volumes: This equals the cube of the scale factor (k³)

For example, if the scale factor is 2 (one figure is twice as large as the other):

  • Corresponding sides: ratio of 2:1
  • Perimeters: ratio of 2:1
  • Areas: ratio of 4:1 (2² = 4)
  • Volumes: ratio of 8:1 (2³ = 8)

This relationship exists because:

  • Perimeter involves 1-dimensional measurements (linear)
  • Area involves 2-dimensional measurements (squared)
  • Volume involves 3-dimensional measurements (cubed)

Always pay attention to what you're comparing to use the correct ratio!

Question: What properties of similar figures stay the same? What changes?

Answer: In similar figures, the following properties STAY THE SAME:

  • Shape of the figure
  • All angle measures
  • Parallelism (parallel lines remain parallel)
  • Perpendicularity (perpendicular lines remain perpendicular)
  • Ratio of corresponding linear measurements
  • Basic geometric relationships

What CHANGES:

  • Size of the figure (all lengths change by the scale factor)
  • Distance from any point to a reference point
  • Perimeter (changes by scale factor)
  • Area (changes by square of scale factor)
  • Volume (changes by cube of scale factor)

This is why similar figures have the same shape but different sizes. The geometric relationships remain consistent while the absolute measurements change proportionally.

For example, if you have two similar triangles, their corresponding angles are identical, but one triangle's sides are all longer or shorter than the other's by the same factor.

Question: Where are similar figures used in real life?

Answer: Similar figures have many practical applications:

  1. Maps and Blueprints: Creating scaled representations of real objects
  2. Photography: Zooming, resizing, and scaling images
  3. Architecture: Building models and creating construction plans
  4. Engineering: Designing parts that need to be scaled up or down
  5. Medicine: Enlarging medical images for diagnosis
  6. Cartography: Creating maps at different scales
  7. Computer Graphics: Resizing and scaling objects in design software

When you see a map with a scale like "1 inch = 10 miles," that's a practical application of similar figures! The mathematical concept models this common real-world scaling process.

In manufacturing, parts are often designed at one scale and then produced at a different scale using similarity principles.

Question: What's the difference between congruent and similar figures?

Answer: The key differences are:

Property Congruent Figures Similar Figures
Shape Same Same
Size Same Different
Corresponding Sides Equal Proportional
Corresponding Angles Equal Equal
Scale Factor 1 Any positive number

All congruent figures are similar (with scale factor 1), but not all similar figures are congruent.

Think of it this way: congruent figures are like identical twins (same size and shape), while similar figures are like parent and child who look alike but are different sizes.

Congruent figures can be mapped onto each other using rigid transformations (translations, rotations, reflections), while similar figures require a dilation as well.