Similarity Formula

The Formula

\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k (scale factor)

When to use: A photo and its enlargement are similar—same shape, different size.

Quick Example

A 3-4-5 triangle is similar to a 6-8-10 triangle (scale factor 2).

Notation

\sim means 'is similar to'

What This Formula Means

Two figures are similar if they have the same shape but possibly different sizes.

A photo and its enlargement are similar—same shape, different size.

Formal View

F_1 \sim F_2 \iff \exists similarity transformation T (isometry \circ dilation) with T(F_1) = F_2; equivalently \exists\, k > 0 such that \forall P,Q \in F_1: |T(P)T(Q)| = k\,|PQ|

Worked Examples

Example 1

medium
Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, AC = 10, and DE = 9, find EF and DF.

Solution

  1. 1
    Find the scale factor: k = \frac{DE}{AB} = \frac{9}{6} = 1.5.
  2. 2
    Multiply each corresponding side by the scale factor: EF = BC \times 1.5 = 8 \times 1.5 = 12.
  3. 3
    DF = AC \times 1.5 = 10 \times 1.5 = 15.

Answer

EF = 12, \quad DF = 15
Similar figures have equal corresponding angles and proportional corresponding sides. The ratio between any pair of corresponding sides is constant (the scale factor).

Example 2

hard
A tree casts a 15 m shadow at the same time a 2 m pole casts a 3 m shadow. How tall is the tree?

Common Mistakes

  • Confusing with congruence
  • Incorrect scale factor application

Why This Formula Matters

Basis for scale drawings, maps, and proportional reasoning in geometry.

Frequently Asked Questions

What is the Similarity formula?

Two figures are similar if they have the same shape but possibly different sizes.

How do you use the Similarity formula?

A photo and its enlargement are similar—same shape, different size.

What do the symbols mean in the Similarity formula?

\sim means 'is similar to'

Why is the Similarity formula important in Math?

Basis for scale drawings, maps, and proportional reasoning in geometry.

What do students get wrong about Similarity?

Students confuse similar with congruent. Similar shapes have the same shape but can differ in size. All circles are similar; not all rectangles are.

What should I learn before the Similarity formula?

Before studying the Similarity formula, you should understand: congruence, ratios.

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