Geometric Similarity
Two geometric figures are similar if they have the same shape but not necessarily the same size, meaning one can be obtained from the other by a combination of rigid motions and a uniform scaling (dilation). Similar figures have equal corresponding angles and proportional corresponding sides. Similarity is denoted by the symbol ~ and is the basis for scale drawings, maps, and many real-world applications including shadow calculations and indirect measurement.
Key Formula
a1/a2 = b1/b2 = c1/c2 = k (scale factor)
LaTeX: \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k
| Symbol | Meaning | Unit |
|---|---|---|
| a1, b1, c1 | side lengths of the first figure | units |
| a2, b2, c2 | corresponding side lengths of the second figure | units |
| k | scale factor (ratio of similarity) | unitless |
Worked Example
Problem
Triangle ABC has sides 6 cm, 8 cm, and 10 cm. Triangle DEF is similar to ABC with a scale factor of 1.5. Find the sides of triangle DEF.
Solution
Step 1: Scale factor k = 1.5. Step 2: Side DE = 6 × 1.5 = 9 cm. Step 3: Side EF = 8 × 1.5 = 12 cm. Step 4: Side DF = 10 × 1.5 = 15 cm.
Answer
Triangle DEF has sides 9 cm, 12 cm, and 15 cm
Triangle Similarity Criteria
| Criterion | Full Name | Required Condition | Notes |
|---|---|---|---|
| AA | Angle-Angle | 2 corresponding angles are equal | Third angle automatically equal (angle sum = 180°) |
| SSS | Side-Side-Side (ratio) | All 3 sides in same ratio | Scale factor must be consistent |
| SAS | Side-Angle-Side (ratio) | 2 sides in ratio, included angle equal | Most common indirect measurement method |
Interactive Tools
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Related Terms
Geometric Congruence
Two geometric figures are congruent if they have exactly the same shape and size, meaning one can be transformed into the other through rigid motions such as translation, rotation, or reflection without any stretching or scaling. Congruence is denoted by the symbol ≅ and is a foundational concept for proving geometric theorems and properties. It is distinct from similarity, which allows size differences while preserving shape.
Triangle
A triangle is a polygon with exactly three sides, three angles, and three vertices. The sum of the interior angles of any triangle always equals 180°, making it the simplest closed polygon. Triangles are the most rigid of all polygons and are widely used in engineering structures, architecture, and navigation due to their inherent stability.
Pythagorean Theorem
The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (the legs). It is one of the most famous and widely applied theorems in mathematics, used in distance calculations, navigation, construction, and virtually every branch of science and engineering.
From Latin "similis" meaning like or resembling, related to "semel" (once, same). The suffix "-ity" comes from Latin "-itas" denoting a state or quality. The mathematical use of "similar" to describe proportional figures was formalized by Euclid in "Elements" around 300 BC.