Same angles, sides in the same ratio. Congruence is similarity with a scale factor of 1.
Find the side
Triangle ABC is similar to triangle DEF. If AB = 6, DE = 9 and BC = 8, how long is EF?
The scale factor is 96 = 1.5, so every side of DEF is 1.5 times its partner: EF = 8 × 1.5 = 12. A adds the difference (9 − 6 = 3, so 8 + 3 = 11) — similarity scales by multiplication, not addition. C divides instead of multiplying, shrinking the triangle that should have grown.
Angles equal, corresponding sides in the same ratio.
The symbol is ~ for similar and ≅ for congruent. Congruent triangles are similar with scale factor 1 — everything true of similar triangles is true of congruent ones.
| Test | What you need |
|---|---|
| AA | two pairs of equal angles |
| SSS similarity | all three pairs of sides in the same ratio |
| SAS similarity | two pairs in ratio, with equal included angles |
AA needs only two, because the third follows from the angle sum. This is the test that fails for congruence — AAA fixes the shape but not the size, which is exactly what similarity asks for.
△ABC ~ △DEF
A↔D, B↔E, C↔F
AB/DE = BC/EF
small over large, both sides
69 = 8EF
substitute
6 · EF = 72, EF = 12
cross multiply
Writing AB/DE on one side and EF/BC on the other mixes the direction and gives a wrong answer that still looks tidy. Decide which triangle is the numerator and stay with it.
Area involves two lengths multiplied, so the scale factor is applied twice. A photograph enlarged to twice the width uses four times the paper.
A 1.6 m person casts a 2 m shadow
a tree casts 15 m
1.62 = h15
same sun angle → AA similarity
2h = 24, h = 12 m
the tree's height
The two triangles are similar because the sun's rays arrive at the same angle and both objects stand upright — two equal angles, which is all AA requires. This is how heights were measured long before anyone could climb them.
Basic proportionality theorem: a line parallel to one side of a triangle cuts the other two sides in the same ratio.
In △ABC, DE ∥ BC with D on AB and E on AC. AD = 4, DB = 6 and AE = 6. Find EC.
AD/DB = AE/EC
piece over piece on both sides
46 = 6EC
substitute
4 · EC = 36, EC = 9
check: 4 : 6 = 6 : 9 ✓
Writing AD/AB = AE/EC mixes a piece with a whole side: 410 = 6EC gives EC = 15. Keep it piece over piece (AD/DB = AE/EC) or whole over whole (AD/AB = AE/AC).
Similar triangles have perimeters 12 cm and 30 cm. Find the ratio of their areas.
12 : 30 = 2 : 5
perimeters scale like sides
areas 2² : 5² = 4 : 25
square the scale factor
The altitude to the hypotenuse of a right triangle splits it into pieces of 4 and 9. How long is the altitude?
all three triangles are similar
each shares an angle with the original (AA)
4h = h9
matching sides of the two small triangles
h² = 36, h = 6
the altitude is the geometric mean
Scale factors below 1, square roots, and when the whole side belongs in the proportion.
△ABC ~ △XYZ with AB = 15, XY = 6 and BC = 10. Find YZ.
k = 615 = 25
going to the smaller triangle
YZ = 10 × 25 = 4
check: 15 : 10 = 6 : 4 ✓
A scale factor below 1 shrinks; the answer must be smaller than 10.
Similar triangles have areas 18 cm² and 50 cm². Find the ratio of their sides.
18 : 50 = 9 : 25
simplify first
sides √9 : √25 = 3 : 5
undo the square with a root
From areas to sides is the square root; from sides to areas is the square.
In △ABC, DE ∥ BC, AD = 3, AB = 9 and DE = 5. Find BC.
△ADE ~ △ABC
AA: corresponding angles
AD/AB = DE/BC, so 39 = 5BC
whole sides of the two triangles
BC = 15
check: 3 : 9 = 5 : 15 ✓
DE and BC are sides of the small and large triangles, so AB (the whole) is correct here, not DB.
Expect a missing side from two similar triangles, a shadow or height problem, a line parallel to one side of a triangle, and a ratio of areas from a ratio of sides or perimeters. The fastest route is the scale factor: match the letters, divide one pair of corresponding sides, then multiply — and square it for areas. The 10-second check: the answer must grow if the triangle grows and shrink if it shrinks, and both ratios in your proportion must simplify to the same fraction.
Find the area
Two similar triangles have sides in the ratio 2 : 5. The smaller has an area of 12 cm². What is the area of the larger?
Areas scale by the square of the side ratio: (52)² = 254 = 6.25, so the larger area is 12 × 6.25 = 75 cm². A uses the side ratio directly (12 × 2.5 = 30), which would be right for a length but not an area. Check: 7512 = 6.25 = (52)² ✓
Hint: square the ratio before multiplying. An area involves two lengths, so the scale factor applies twice.
