🔍 Similar Figures: The Secret Language of Shapes
Imagine you have a tiny photo of your family. Now imagine blowing it up to poster size. Everyone looks the same—just bigger! That’s the magic of similar figures.
🌟 The Big Idea
Similar figures are shapes that look exactly alike, just different sizes. Like a baby elephant and its parent—same shape, different size!
Think of it like a photocopier with a zoom button. You can make things bigger or smaller, but the shape stays the same.
📐 Ratio and Proportion
What’s a Ratio?
A ratio compares two numbers. It’s like saying:
“For every 1 apple, I have 2 oranges”
We write this as 1:2 or 1/2.
Simple Example:
- You have 3 red balls and 6 blue balls
- The ratio is 3:6, which simplifies to 1:2
- This means for every 1 red ball, there are 2 blue balls!
What’s a Proportion?
A proportion says two ratios are equal.
1/2 = 2/4 = 3/6
All of these mean the same thing! Half is half, no matter how you write it.
Real Life Example:
- A recipe for 2 people needs 1 cup of rice
- For 4 people? Just double it: 2 cups!
- The ratio stays the same: 1:2 = 2:4
🔺 Similar Triangles Introduction
Meet the Similar Triangles
Two triangles are similar when:
- ✅ They have the same angles
- ✅ Their sides are proportional (same ratio)
Imagine This:
- You draw a triangle on paper
- Your friend takes a photo and zooms in 2x
- Both triangles look the same, just different sizes
- These are similar triangles!
graph TD A["Small Triangle 🔺"] --> B["Same Angles"] C["Big Triangle 🔺"] --> B B --> D["Similar! ✓"]
Key Point: Similar triangles have the exact same shape. Only the size changes!
✨ Triangle Similarity Criteria
How do we know if two triangles are similar? We have three easy tests!
1. AA (Angle-Angle)
If two angles match, the triangles are similar.
Why only two? Because triangle angles always add to 180°. If two match, the third must match too!
Example:
- Triangle A has angles: 30°, 60°, 90°
- Triangle B has angles: 30°, 60°, 90°
- ✅ Same angles = Similar!
2. SSS (Side-Side-Side)
If all three sides have the same ratio, they’re similar.
Example:
- Triangle A sides: 3, 4, 5
- Triangle B sides: 6, 8, 10
- Check ratios: 3/6 = 4/8 = 5/10 = 1/2
- ✅ Same ratio = Similar!
3. SAS (Side-Angle-Side)
If two sides have the same ratio AND the angle between them matches, they’re similar.
Example:
- Triangle A: sides 3 and 4 with 90° between
- Triangle B: sides 6 and 8 with 90° between
- Ratio: 3/6 = 4/8 = 1/2 ✓
- Angle: 90° = 90° ✓
- ✅ Similar!
graph TD A["Two Triangles"] --> B{Which test?} B --> C["AA: Two angles match?"] B --> D["SSS: All sides same ratio?"] B --> E["SAS: Two sides ratio + angle?"] C --> F["✅ Similar!"] D --> F E --> F
🎯 Corresponding Parts
What Does “Corresponding” Mean?
Corresponding parts are the matching pieces between similar figures.
Think of it like matching socks:
- Left sock matches left sock
- Right sock matches right sock
- The small sock’s heel matches the big sock’s heel!
In Similar Triangles:
| Small Triangle | Big Triangle | They Match! |
|---|---|---|
| Angle A | Angle D | Same angle |
| Angle B | Angle E | Same angle |
| Angle C | Angle F | Same angle |
| Side AB | Side DE | Same ratio |
| Side BC | Side EF | Same ratio |
| Side CA | Side FD | Same ratio |
Example:
- Small triangle has sides: 2, 3, 4
- Big triangle has sides: 4, 6, 8
- Side 2 corresponds to side 4 (both are shortest)
- Side 4 corresponds to side 8 (both are longest)
📏 Scale Factor
The Magic Multiplier
The scale factor is the number you multiply by to go from one figure to another.
It’s like the zoom level on your camera!
Formula:
Scale Factor = Big Side ÷ Small Side
Example:
- Small square: 3 cm sides
- Big square: 9 cm sides
- Scale factor = 9 ÷ 3 = 3
- The big square is 3 times larger!
Going Both Ways
| Direction | Scale Factor | What Happens |
|---|---|---|
| Small → Big | Greater than 1 | Enlargement |
| Big → Small | Less than 1 | Reduction |
| Same size | Exactly 1 | No change |
Example:
- Photo is 4 inches
- Poster is 12 inches
- Scale factor (small→big) = 12/4 = 3
- Scale factor (big→small) = 4/12 = 1/3
🗺️ Scale Drawings
Maps and Models
A scale drawing represents something larger (or smaller) using a fixed ratio.
Real Life Examples:
- 🗺️ Maps: 1 inch = 100 miles
- 🏠 House plans: 1 cm = 1 meter
- 🚗 Toy cars: 1:24 scale
How to Read Scale
If a map says 1:1000:
- 1 cm on map = 1000 cm in real life
- 1 cm on map = 10 meters in real life!
Example Problem:
- Map scale: 1 cm = 5 km
- Distance on map: 3 cm
- Real distance = 3 × 5 = 15 km
How to Make Scale Drawings
- Choose your scale (like 1:100)
- Divide real measurements by scale
- Draw with new measurements
Example:
- Real room: 5 meters × 4 meters
- Scale: 1 cm = 1 meter
- Drawing: 5 cm × 4 cm
📏 Similar Perimeter Ratio
Perimeters and Scale Factor
Here’s a cool discovery:
The ratio of perimeters equals the scale factor!
Why? Because perimeter is just adding up the sides. If each side is multiplied by the scale factor, the total is too!
Example:
- Small triangle perimeter: 12 cm
- Scale factor: 2
- Big triangle perimeter: 12 × 2 = 24 cm
- Ratio of perimeters: 24/12 = 2 ✓
graph TD A["Scale Factor = k"] --> B["Each side × k"] B --> C["Perimeter × k too!"] C --> D["Perimeter Ratio = k"]
Another Example:
- Rectangle A: sides 2 and 3, perimeter = 10
- Rectangle B: sides 4 and 6, perimeter = 20
- Scale factor = 4/2 = 2
- Perimeter ratio = 20/10 = 2 ✓
🟦 Area Ratio of Similar Figures
The Square Surprise!
This is where it gets exciting:
The area ratio equals the scale factor SQUARED!
Why? Area uses TWO dimensions (length × width). Each dimension gets multiplied by the scale factor!
Formula:
Area Ratio = (Scale Factor)²
Example:
- Small square: 2 cm × 2 cm = 4 cm²
- Scale factor: 3
- Big square: 6 cm × 6 cm = 36 cm²
- Area ratio: 36/4 = 9
- Check: 3² = 9 ✓
The Pattern
| Scale Factor | Perimeter Ratio | Area Ratio |
|---|---|---|
| 2 | 2 | 4 (2²) |
| 3 | 3 | 9 (3²) |
| 4 | 4 | 16 (4²) |
| 5 | 5 | 25 (5²) |
Real Life Example:
- Small pizza: 8 inch diameter, costs $8
- Large pizza: 16 inch diameter, costs $16
- Scale factor = 16/8 = 2
- Area ratio = 2² = 4
- The large pizza has 4 times more food but only costs 2 times more!
- 🍕 Big pizza = better deal!
graph TD A["Scale Factor = k"] --> B["Perimeter Ratio = k"] A --> C["Area Ratio = k²"] B --> D["Linear relationship"] C --> E["Quadratic relationship"]
🎯 Quick Summary
| Concept | Key Rule | Example |
|---|---|---|
| Ratio | Compare two numbers | 3:6 = 1:2 |
| Proportion | Equal ratios | 1/2 = 3/6 |
| Similar Figures | Same shape, different size | Photo → Poster |
| AA Test | 2 angles match | 30°, 60° |
| SSS Test | All sides same ratio | 3,4,5 and 6,8,10 |
| SAS Test | 2 sides ratio + angle | Works too! |
| Corresponding | Matching parts | Shortest to shortest |
| Scale Factor | The multiplier | k = 3 means 3× bigger |
| Scale Drawing | Real thing made smaller | Maps use this |
| Perimeter Ratio | Equals scale factor | k = 2, perimeter × 2 |
| Area Ratio | Scale factor squared | k = 2, area × 4 |
🌈 Remember This!
Similar figures are like family photos at different sizes:
- 👶 Baby picture (small)
- 🧒 School photo (medium)
- 📸 Portrait (large)
All show the same person, just at different sizes. The proportions stay the same, and that’s what makes them similar!
Now you speak the secret language of shapes. Go find similar figures everywhere—they’re all around you!