Similar figures
Two figures are similar if they have the same shape but not necessarily the same size. All circles are similar; all squares are similar; a photograph and its enlargement are similar.
Two polygons with the same number of sides are similar if:
their corresponding angles are equal, and
their corresponding sides are in the same ratio (proportional).
Both conditions are needed for polygons in general. A square and a rectangle have equal angles but are not similar; a square and a rhombus have proportional sides but are not similar.
Congruent figures are similar figures with a ratio of 1.
The Basic Proportionality Theorem
Theorem (Thales). If a line is drawn parallel to one side of a triangle, intersecting the other two sides at distinct points, it divides those two sides in the same ratio.
In △ABC, if DE ∥ BC with D on AB and E on AC, then
AD/DB = AE/EC
Idea of the proof. Triangles BDE and DEC lie on the same base DE and between the same parallels DE and BC, so they have equal areas. Comparing the areas of △ADE with each of them (using ½ × base × height with a shared height) gives AD/DB = area(ADE)/area(BDE) = area(ADE)/area(DEC) = AE/EC.
Example. In △ABC, DE ∥ BC, AD = 3 cm, DB = 6 cm and AE = 2 cm. Find EC.
3/6 = 2/EC, so EC = 4 cm.
Converse. If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
Criteria for similarity of triangles
For triangles, you don't have to check everything. Any one of these is enough:
Criterion | Condition | Then |
|---|---|---|
AAA / AA | two angles of one triangle equal two angles of the other (the third angles are then equal too) | the triangles are similar |
SSS | all three pairs of sides are in the same ratio | the triangles are similar |
SAS | one angle is equal and the sides including it are in the same ratio | the triangles are similar |
If △ABC ~ △PQR (the symbol ~ means "is similar to"), the vertex order shows the matching, and
AB/PQ = BC/QR = CA/RP, with ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R.
For triangles alone, equal angles guarantee proportional sides and vice versa. That is why AA is enough.
Using similarity
Example: shadows. A vertical pole 6 m tall casts a shadow 4 m long. At the same time, a tower casts a shadow 28 m long. How tall is the tower?
The Sun's rays make the same angle with the ground for both, and both stand vertically, so the triangles formed are similar (AA). Height/shadow is the same: 6/4 = h/28, so h = 42 m.
Example: a street lamp. A boy 1.5 m tall stands 6 m from the foot of a lamp post 4.5 m high. Let his shadow be x m long. The lamp post and the boy are both vertical, so the triangles are similar: 4.5/(6 + x) = 1.5/x. Then 4.5x = 1.5(6 + x), so 3x = 9 and x = 3 m.
Example: proving similarity. In △ABC, ∠ACB = 90° and CD ⊥ AB. Then △ADC ~ △ACB (they share ∠A and both have a right angle, AA), and △CDB ~ △ACB similarly. From △ADC ~ △ACB, AC/AB = AD/AC, so AC² = AB × AD. Results like this lead to a proof of Pythagoras' theorem.
Pythagoras through similarity
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Using the result above for both legs:
AC² = AB × AD and BC² = AB × BD, so AC² + BC² = AB × (AD + BD) = AB².
The converse also holds: if the square of one side equals the sum of the squares of the other two, the angle opposite that side is a right angle.
Common mistakes
Matching the wrong sides. Always follow the vertex order in △ABC ~ △PQR.
Using whole sides in the BPT. AD/DB = AE/EC uses the parts; AD/AB = AE/AC is also true, but mixing the two (AD/AB = AE/EC) is wrong.
Applying SAS with a non-included angle. The equal angle must lie between the proportional sides.
Summary
Similar figures have the same shape: equal angles and proportional sides.
BPT: a line parallel to one side divides the other two sides in the same ratio; its converse also holds.
Triangles are similar by AA, SSS or SAS.
Similarity solves height-and-shadow problems and proves Pythagoras' theorem.