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. Here you can resize a figure to see how its sides and area change with the ratio.

Intuition — Same as Zooming a Photo

Similarity Ratio1.5

📱 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!

Angle A50°

Angle B70°

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

④In practice 90% of problems use AA

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.

The ratio of corresponding sides equals the similarity ratio.

2 : 3 = 4 : x

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.

The area ratio is the square of the similarity ratio.

area ratio = 2² : 3²

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

The area ratio is the square of the similarity ratio: 1² : 2² = 1 : 4.

area ratio = 1 : 4

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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