Similarity is the relationship between two figures with exactly the same shape but different sizes.
Just as a face keeps its features when a photo is enlarged or shrunk, similar figures have all corresponding angles equal and all corresponding sides in the same ratio.
That fixed ratio is the scale factor, and because doubling the lengths makes areas four times and volumes eight times as large, the ratio of areas is the square of the scale factor and the ratio of volumes is its cube.
For triangles in particular, two equal angles force the third angle to be equal as well, so checking two angles is enough to decide similarity.
Enlarging and shrinking
seegongsik.com
Triangle A'B'C' growing and shrinking with the scale factor, and its side and area ratios to triangle ABC
1.5
📱 Similarity through photos
①Spreading two fingers on a phone photo keeps the shape and only makes it larger: that is similarity.
②In two similar figures, all corresponding angles are equal.
③All corresponding sides are in the same ratio, and that ratio is the scale factor.
④With k = 1.5 in the picture, the side AB = 80 becomes A'B' = 120, and the other two sides are also multiplied by 1.5.
Two equal angles mean similar
seegongsik.com
A small and a large triangle given the same two angles, and the third angle that becomes equal by itself
50°
70°
Conditions for similar triangles
SSS similarity · SAS similarity · AA similarity
Of the three, AA similarity is the one used most in exams.
💡 The three similarity conditions
①SSS: if all three pairs of corresponding sides are in the same ratio, the triangles are similar.
②SAS: if two pairs of corresponding sides are in the same ratio and the angles between them are equal, the triangles are similar.
③AA: if two pairs of corresponding angles are equal, the triangles are similar. Since the angles add up to 180°, the remaining angle is automatically equal.
④Changing ∠A and ∠B in the picture shows the third angle of the two different-sized triangles always following with the same value.
Scale factor, area ratio and volume ratio
Scale factor and area ratio
scale factor m : n → area ratio m² : n²
Area is length times length, so its ratio is squared.
Scale factor and volume ratio
scale factor m : n → volume ratio m³ : n³
Volume is a product of three lengths, so its ratio is cubed.
📝 Why squares and cubes?
①Doubling the side of a square doubles both its width and its height, so its area becomes 2 × 2 = 4 times as large.
②Doubling the edge of a cube doubles width, depth and height, so its volume becomes 2 × 2 × 2 = 8 times as large.
③So a scale factor of 1 : 2 gives an area ratio of 1 : 4 and a volume ratio of 1 : 8.
Similarity in everyday life
🌳 Measuring a tree with shadows
①At the same moment, measure the length of your shadow and the tree's shadow.
②Sunlight arrives in parallel rays, so the triangle made by you, your shadow and the light ray is AA-similar to the one made by the tree, its shadow and the light ray.