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Scale factor in similar triangles

The scale factor is the number by which one similar triangle is enlarged or reduced to get the other.

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Student-friendly explanation

Scale factor shows the size change between similar triangles. If the factor is k, then each corresponding side is multiplied by k, while area changes by k².

How to write this in exams

  1. 1

    Start with the exact idea

    The scale factor is the number by which one similar triangle is enlarged or reduced to get the other.

  2. 2

    Then show how to use it

    1. Identify the direction of scaling. 2. Multiply or divide corresponding sides by the factor. 3. For area, square the factor. 4. Check that the order of triangles is correct.

  3. 3

    Add one concrete example

    If a triangle is enlarged by scale factor 3, a 4 cm side becomes 12 cm.

  4. 4

    Avoid this incomplete answer

    Using the same factor for area and sides. Area must be multiplied by the square of the scale factor.

Definition

The scale factor is the number by which one similar triangle is enlarged or reduced to get the other.

Example

If a triangle is enlarged by scale factor 3, a 4 cm side becomes 12 cm.

Rule to remember

Side ratio = scale factor; perimeter ratio = scale factor; area ratio = (scale factor)^2.

Memory hook

Sides by k, areas by k².

Examples and method

Worked example

If the scale factor from a small triangle to a large triangle is 4, then a 3 cm side becomes 12 cm. The area would become 16 times larger because 4² = 16.

Method to apply

1. Identify the direction of scaling. 2. Multiply or divide corresponding sides by the factor. 3. For area, square the factor. 4. Check that the order of triangles is correct.

Diagram support

Write the smaller and larger triangles in the same vertex order and mark matching sides.

How CBSE asks it

A question may ask for enlarged side lengths, reduced side lengths, or the factor from one triangle to another.

Avoid common mistakes

Common confusion

Students sometimes reverse the factor or use it directly for area without squaring.

Common wrong answer

Using the same factor for area and sides. Area must be multiplied by the square of the scale factor.

Exam tip

Decide first whether you are moving from the smaller triangle to the larger one or the other way around.

Quick check

What does the scale factor tell you in two similar triangles?

The scale factor tells how much one similar triangle is multiplied to get the other. It applies directly to sides, and the area changes by the square of that factor.

Answer writing and exam use

1-mark answer

The scale factor is the number by which one similar triangle is enlarged or reduced to get the other.

2-mark answer

The scale factor is the number by which one similar triangle is enlarged or reduced to get the other. Side ratio = scale factor; perimeter ratio = scale factor; area ratio = (scale factor)^2. If a triangle is enlarged by scale factor 3, a 4 cm side becomes 12 cm.

3-mark answer

Scale factor shows the size change between similar triangles. If the factor is k, then each corresponding side is multiplied by k, while area changes by k². Side ratio = scale factor; perimeter ratio = scale factor; area ratio = (scale factor)^2. If the scale factor from a small triangle to a large triangle is 4, then a 3 cm side becomes 12 cm. The area would become 16 times larger because 4² = 16. A question may ask for enlarged side lengths, reduced side lengths, or the factor from one triangle to another. Using the same factor for area and sides. Area must be multiplied by the square of the scale factor.
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