Lesson: Similarity - Grade 8 (M.2)
Hello to all my Grade 8 students! š
If you feel like math is a bit tricky at first, don't worry at all! Today, we're going to get to know the concept of "Similarity," which is a fun topic that is much more relevant to our daily lives than you might think. I'll show you how different shapes can be "similar" and how we can use this knowledge to calculate the height of a building or a tree without having to climb up to measure them!
1. What is Similarity? (Easy to understand in 1 minute)
Imagine when you use your phone to "Zoom in" or "Zoom out" on a photo. The image looks exactly the same, but the size changes. This is the heart of similarity!
Key point: Two similar figures don't need to be the same size (that's called congruence), but they must have the same "shape."
Did you know?
The maps we use for navigation or toy car models are all created using the principle of similarity, ensuring that the proportions between the real object and the miniature remain exactly the same!
2. Similar Polygons
For two polygons to be similar, they must satisfy 2 criteria (both must be met!):
1. Corresponding angles are equal in measure: Angle by angle, they must be identical.
2. The ratios of the lengths of corresponding sides are equal: If one side is enlarged by a factor of 2, every other side must also be enlarged by a factor of 2.
Symbol: We use the symbol \( \sim \) to represent "is similar to."
Example: Quadrilateral \( ABCD \sim \) Quadrilateral \( PQRS \)
Common Mistake: Having only equal sides or only equal angles (except for triangles) is not enough to say that shapes are similar. For example, a square and a rectangle both have all 90-degree angles, but they are not similar!
3. The Star of the Show: Similar Triangles
For triangles, your life gets much easier! Because we only need to check the "angles."
Theorem: Two triangles are similar if and only if "they have 3 pairs of equal corresponding angles."
Memory Tip:
If two pairs of angles are equal, the third pair will be equal automatically (because the sum of interior angles in a triangle must be 180 degrees). So, if you see just 2 pairs of equal angles, you can conclude that they are similar!
4. Finding Missing Side Lengths (Step-by-Step)
Once we know that two triangles are similar, we can find unknown side lengths using "ratios."
Steps:
1. Match corresponding sides: The sides opposite the equal angles are the corresponding sides.
2. Set up the ratio equation: Take the side length of the larger figure and divide it by the corresponding side length of the smaller figure (or vice versa, but be consistent for every pair).
3. Solve the equation: Find the value of the variable you need.
Example:
If triangle \( ABC \sim \) triangle \( DEF \),
The ratio will be: \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \)
Suppose: \( AB = 4 \), \( DE = 8 \), and \( BC = 5 \). Find \( EF \).
The equation will be: \( \frac{4}{8} = \frac{5}{EF} \)
Cross multiply: \( 4 \times EF = 40 \)
Therefore, \( EF = 10 \)
5. Word Problems and Applications
One of the most popular problems is finding the height of a tree using its shadow.
Scenario: You are 150 cm tall and standing near a tree. You cast a shadow that is 200 cm long. The tree casts a shadow that is 8 meters (800 cm) long. How tall is the tree?
Approach: Sunlight hitting at the same angle creates similar triangles.
- Let \( x \) be the height of the tree.
- Set up the proportion: \( \frac{\text{Tree Height}}{\text{Person Height}} = \frac{\text{Tree Shadow}}{\text{Person Shadow}} \)
- \( \frac{x}{150} = \frac{800}{200} \)
- \( \frac{x}{150} = 4 \)
- \( x = 4 \times 150 = 600 \) cm, or 6 meters!
6. Common Errors (Checklist!)
ā¢ Incorrect order of vertices: When writing \( \triangle ABC \sim \triangle DEF \), always ensure the order of the angles matches. Angle \( A \) must correspond with \( D \), \( B \) with \( E \), and so on.
ā¢ Confusing similarity with congruence: Remember, "congruent" means identical twins (same shape, same size), but "similarity" is like a zoomed photo (same shape, different size).
ā¢ Swapping numbers: If you put the larger shape's side in the numerator, do it for every fraction. Never flip the top and bottom values randomly.
Key Takeaways
ā¢ Similar figures have the same shape but may have different sizes.
ā¢ Similar polygons must have equal angles and proportional sides.
ā¢ Similar triangles only require checking that "3 pairs of angles are equal."
ā¢ Benefits: It is used to calculate distances or heights that are difficult to measure directly.
Keep it up, everyone! If you visualize it as "zooming," you'll definitely find these problems fun to solve. Practice regularly, and you'll see it's not that hard at all! āļø