Similar Figure
Two figures are said to be similar if they have the exact same shape but not necessarily the same size. For polygons with the same number of sides to be similar, their corresponding angles must be equal and the lengths of their corresponding sides must be proportional.
Key Concepts
All congruent figures are similar because they have the same shape and size, but similar figures are not always congruent.
The constant ratio of the corresponding sides of similar figures is known as the scale factor.
For triangles, specific similarity criteria such as AAA (Angle-Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side) are used to prove similarity.
Formula & Equation
Triangle ABC is similar to Triangle DEF if Angle A = Angle D, Angle B = Angle E, Angle C = Angle F, and AB/DE = BC/EF = CA/FD.
ABC and DEF are the vertices of the two triangles. The angles represent corresponding angles, and the fractions represent the ratios of corresponding side lengths.
Common Misconceptions
Myth: If two polygons have proportional sides, they are automatically similar.
Fact: This is only true for triangles. For polygons with four or more sides, like a square and a rhombus, having proportional sides is not enough; their corresponding angles must also be equal.
Real World Applications
A scale model of a car and the actual life-size car are similar figures.
A photograph printed in both passport size and a large poster size represents similar figures.
Frequently Asked Questions
What is the difference between similar and congruent figures?
Congruent figures have the exact same shape and size, while similar figures have the same shape but can have different sizes.
Are all equilateral triangles similar?
Yes, all equilateral triangles are similar because their corresponding angles are always equal to 60 degrees and their sides are always proportional.