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Grade 9 · lesson 3 of 4

Scale Corresponding Lengths

Use one scale factor for corresponding sides of similar figures.

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Supplied materials

  • Paper or a note app.
  • Supplied similar right triangles: original sides 5, 12, 13 units; enlarged sides have scale factor 2.

Learn the idea

Similar figures have equal corresponding angles and proportional corresponding side lengths. A dilation multiplies every length by the same positive scale factor. Doubling only one side would generally change the shape rather than produce a similar triangle.

To recover a scale factor, divide a new side by its corresponding original side. Then apply that factor to the other lengths. Keep correspondences labeled: compare a short leg with a short leg, not with a hypotenuse.

  1. Label the 5 and 12 sides as legs and 13 as hypotenuse.
  2. Multiply each by 2 to get 10, 24, 26.
  3. Check 10² + 24² = 26², then infer a factor from the supplied 6-to-15 correspondence below.
Worked example

See the method in action

The enlarged triangle has sides 10, 24, 26. Ratios 10/5, 24/12, and 26/13 all equal 2. Also 100 + 576 = 676 = 26², so the right-angle relationship is preserved.

Your turn

Check your reasoning

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A 6-8-10 right triangle is enlarged so its 6-unit leg becomes 15. What is the corresponding length of the 8-unit leg?

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See the answer and explanation

20 units

The factor is 15 ÷ 6 = 2.5. Multiplying the corresponding 8-unit leg gives 8 × 2.5 = 20 units. The hypotenuse would be 25.

Take it further

Scale the original 6-8-10 triangle by 1/2 instead. Its sides become 3, 4, 5. A positive factor below one shrinks every length while retaining similarity.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.