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medium difficulty5 min

Scale factor in similar triangles

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

In one line: Sides by k, areas by k².

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Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1If the scale factor from a small triangle to a large triangle is 4, then a 3 cm side becomes 12 cm.
  2. 2The area would become 16 times larger because 4² = 16.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
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.

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

A question may ask for enlarged side lengths, reduced side lengths, or the factor from one triangle to another.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    Scale factor shows the size change between similar triangles.

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