Similarity Formula

Similarity is two figures are similar if they have the same shape but possibly different sizes, meaning all corresponding angles are equal and all.

The Formula

aa′=bb′=cc′=k\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 33-44-55 triangle is similar to a 66-88-1010 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, meaning all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).

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

Formal View

F1∼F2  ⟺  ∃F_1 \sim F_2 \iff \exists similarity transformation TT (isometry ∘\circ dilation) with T(F1)=F2T(F_1) = F_2; equivalently ∃ k>0\exists\, k > 0 such that ∀P,Q∈F1:∣T(P)T(Q)∣=k ∣PQ∣\forall P,Q \in F_1: |T(P)T(Q)| = k\,|PQ|

Worked Examples

Example 1

medium
Triangle ABCABC is similar to triangle DEFDEF. If AB=6AB = 6, BC=8BC = 8, AC=10AC = 10, and DE=9DE = 9, find EFEF and DFDF.

Answer

EF=12,DF=15EF = 12, \quad DF = 15

First step

1
Find the scale factor: k=DEAB=96=1.5k = \frac{DE}{AB} = \frac{9}{6} = 1.5.

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Example 2

hard
A tree casts a 1515 m shadow at the same time a 22 m pole casts a 33 m shadow. How tall is the tree?

Example 3

medium
In △ABC\triangle ABC, a line through DD on ABAB parallel to BCBC meets ACAC at EE. If AD=3,DB=6AD=3, DB=6, AE=4AE=4, find ECEC.

Common Mistakes

  • Expecting similar figures to have equal sides — sides are proportional by a scale factor, not equal.
  • Setting up the proportion with mismatched corresponding sides — pair each side with its true counterpart.
  • Treating same size as required — similar figures can be any size as long as the shape and angles match.

Why This Formula Matters

Similarity formalizes 'scaled copy' and powers scale drawings, maps, and indirect measurement — it lets you find an unknown length (a tree's height) from a known proportion, which congruence's equal-sides rule cannot do. Recognizing it by "Are all corresponding angles equal and all corresponding sides in the same ratio?" — rather than by familiar numbers — is what lets a student tell it apart from congruence and scale drawings and ratio / proportion in a mixed problem set.

Frequently Asked Questions

What is the Similarity formula?

Two figures are similar if they have the same shape but possibly different sizes, meaning all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).

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?

Similarity formalizes 'scaled copy' and powers scale drawings, maps, and indirect measurement — it lets you find an unknown length (a tree's height) from a known proportion, which congruence's equal-sides rule cannot do. Recognizing it by "Are all corresponding angles equal and all corresponding sides in the same ratio?" — rather than by familiar numbers — is what lets a student tell it apart from congruence and scale drawings and ratio / proportion in a mixed problem set.

What do students get wrong about Similarity?

The procedure for similarity is the easy part; the trap is expecting similar figures to have equal sides. Asking "Are all corresponding angles equal and all corresponding sides in the same ratio?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Similarity formula?

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

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