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Grade 8 / Middle 2 (age 13-14)

grade 8 Similarity of Figures

Similarity of Figures

Similarity describes two figures with the same shape but different sizes. Just as zooming a photo keeps a person's proportions, similar figures keep a constant ratio between corresponding sides and equal angles. This is the similarity ratio; areas scale as its square and volumes as its cube. Change the size and the angles and the sides and the area follow the ratio.

Intuition — Same as Zooming a Photo

1.5
📱 Similarity via Photos
①Zoom into a photo: shape stays, size changes
②That is similarity!
③Condition: corresponding angles equal
④Corresponding side lengths in equal ratio
⑤Zoom factor = similarity ratio

AA Similarity — Two Angles Suffice!

50°
70°
3 Triangle Similarity Conditions
AA, SAS, SSS
AA shows up the most on tests
💡 Conditions Explained
①AA: two pairs of equal angles → similar (third angle is forced!)
②SAS: two sides in ratio + included angle equal
③SSS: all three sides in equal ratio
④AA is the condition used most often in exams

Length, Area, Volume Ratios

Ratio → Area Ratio
Ratio m : n → Area m² : n²
Area is 2D, so the ratio gets squared
Ratio → Volume Ratio
Ratio m : n → Volume m³ : n³
Volume is 3D, so the ratio gets cubed
📝 Why Squared and Cubed?
①Double a square's side → area × 4
②Double a cube's side → volume × 8
③Area = length × length, volume = length³
④Ratio 1:2 → area 1:4, volume 1:8

Real-life Uses

🌳 Tree Height from Shadow
①You and a tree cast shadows at the same time
②Sun rays are parallel → similar triangles
③Your height : your shadow = tree height : tree shadow
④Set up a proportion to solve
🗺️ Map Scale is Similarity
①Scale 1:50000 = similarity ratio 1:50000
②2 cm on map = 2 × 50000 = 1 km in reality
③Model building 10 cm, scale 1:100 → real 10 m

Work It Out

Example 1
Two similar triangles have similarity ratio 2 : 3. A side of the smaller is 4. Find the corresponding side of the larger.
1
The ratio of corresponding sides equals the similarity ratio.
2 : 3 = 4 : x
2
Solve the proportion.
2x = 12 ⇒ x = 6
▸ 6
Corresponding sides of similar figures share a constant ratio (the similarity ratio).
Example 2
Find the area ratio of two figures with similarity ratio 2 : 3.
1
The area ratio is the square of the similarity ratio.
area ratio = 2² : 3²
2
Compute.
= 4 : 9
▸ 4 : 9
If the similarity ratio is m : n, the area ratio is m² : n².

Exam Wrap-up

Core Formula
AA → side ratio = similarity ratio → area ratio = ratio²
AA is the fastest, most frequently tested approach
Grade-8 school exam type
Two similar triangles have ratio 1 : 2. The smaller has area 5. What is the larger area?
①10
②15
③20
④25
⑤40
▸ ③ 20
1
The area ratio is the square of the similarity ratio: 1² : 2² = 1 : 4.
area ratio = 1 : 4
2
It is 4 times the smaller area.
5 × 4 = 20
🎯 Exam Points
①AA: two equal angles → done (no need for the third)
②Ratio = ratio of corresponding sides
③Area ratio = ratio², volume ratio = ratio³
④Shadow problems → similar triangles
⑤Match correspondence: ABC ~ DEF means A↔D, B↔E, C↔F
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Properties of Quadrilaterals
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Parallel Lines & Proportional Segments
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