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Grade 8 · lesson 2 of 4

Subtract Squares to Recover a Leg

The hypotenuse square equals the sum of leg squares.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A right triangle has hypotenuse 10 cm and one leg 8 cm. Find its other leg. Candidate answers: (A) 164 cm (B) 18 cm (C) 2 cm (D) 6 cm.
  • Independent check inputs: A right triangle has hypotenuse 15 cm and one leg 12 cm. Find its other leg. Candidate answers: (A) 27 cm (B) 3 cm (C) 369 cm (D) 9 cm.

Learn the idea

Dilations change lengths by a common factor and preserve shape. Right-triangle relationships connect squared perpendicular lengths; their converse tests whether supplied side lengths support a right angle, including in a three-dimensional diagonal model. The hypotenuse square equals the sum of leg squares. If one leg is unknown, subtract the known leg’s square from the hypotenuse square before taking the positive square root.

Here is the method on the supplied teaching case. For the missing leg z, z²+(8)²=(10)². Thus z²=36; take the positive root, z=6 cm.

Identify the longest side as the hypotenuse. Adding the squares of a hypotenuse and a leg does not recover the other leg; the final side length uses a positive root and consistent units.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A right triangle has hypotenuse 10 cm and one leg 8 cm. Find its other leg. Candidate answers: (A) 164 cm (B) 18 cm (C) 2 cm (D) 6 cm. Solution: For the missing leg z, z²+(8)²=(10)². Thus z²=36; take the positive root, z=6 cm. The final answer is 6 cm. Identify the longest side as the hypotenuse. Adding the squares of a hypotenuse and a leg does not recover the other leg; the final side length uses a positive root and consistent units.

Your turn

Check your reasoning

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Helpful tips
  • The hypotenuse square equals the sum of leg squares.
  • Identify the longest side as the hypotenuse.
A right triangle has hypotenuse 15 cm and one leg 12 cm. Find its other leg.

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See the answer and explanation

9 cm

For the missing leg z, z²+(12)²=(15)². Thus z²=81; take the positive root, z=9 cm.

Take it further

Changed-input task: A right triangle has hypotenuse 45 cm and one leg 36 cm. Find its other leg. Candidate answers: (A) 27 cm (B) 3321 cm (C) 81 cm (D) 9 cm. Show the mathematical reason, not only the selected answer. Expected result: 27 cm. For the missing leg z, z²+(36)²=(45)². Thus z²=729; take the positive root, z=27 cm.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A right triangle has hypotenuse 20 cm and one leg 16 cm. Find its other leg.
Need a hint?
  • The hypotenuse square equals the sum of leg squares.
  • Identify the longest side as the hypotenuse.
2. A right triangle has hypotenuse 25 cm and one leg 20 cm. Find its other leg.
Need a hint?
  • The hypotenuse square equals the sum of leg squares.
  • Identify the longest side as the hypotenuse.
3. A right triangle has hypotenuse 30 cm and one leg 24 cm. Find its other leg.
Need a hint?
  • The hypotenuse square equals the sum of leg squares.
  • Identify the longest side as the hypotenuse.

Your choices are temporary. Feedback appears after you check all the answers.

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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.