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Geometry · Grade 8 · Study notes

Pythagorean Theorem and Right Triangles

A right triangle has one 90° angle. The Pythagorean theorem connects the lengths of its three sides.

Learning time: 20–25 minutes · Level: Developing

Learning goals

1. Identify the hypotenuse

The hypotenuse is the side opposite the right angle. It is always the longest side. The other two sides are often labelled a and b; the hypotenuse is c.

2. The theorem

For a right triangle, a² + b² = c². Use it only when the triangle is right-angled. Square the two shorter sides, add them, then take the square root to find c.

3. Find a shorter side

If the hypotenuse is known, rearrange: a² = c² − b², so a = √(c² − b²). Choose the positive length in ordinary geometry problems.

4. Check for a right triangle

If the longest side is c, check whether a² + b² = c². If the equality holds, the triangle is right-angled.

5. Apply the theorem

A ladder, diagonal across a rectangular floor, or shortest straight-line distance can form a right triangle. Sketch the situation and label the perpendicular sides before calculating.

Use the theorem and verify the triangle

The Pythagorean theorem applies only to right-angled triangles. The hypotenuse is the side opposite the 90° angle and is always the longest side. If the shorter sides are a and b and the hypotenuse is c, then a² + b² = c². To find a shorter side, rearrange the equation: a² = c² − b².

Worked reasoning: A right triangle has shorter sides 9 cm and 12 cm. Then c² = 9² + 12² = 81 + 144 = 225, so c = 15 cm. Check: 9² + 12² = 15².

The converse can test whether a triangle is right-angled: if the square of its longest side equals the sum of the squares of the other two sides, it is a right triangle. For example, 5, 12 and 13 form one because 25 + 144 = 169.

Worked examples

  1. Sides 3 and 4: c = √(3² + 4²) = √25 = 5.
  2. Hypotenuse 13 and one leg 5: other leg = √(13² − 5²) = √144 = 12.
  3. A rectangle 6 cm by 8 cm has diagonal √(36 + 64) = 10 cm.

Independent practice

Try each question before revealing or checking the worked answer.

  1. Find the hypotenuse when legs are 6 and 8. Answer: √(36 + 64) = 10.
  2. Find the hypotenuse for legs 5 and 12. Answer: 13.
  3. Find the missing leg when c = 10 and one leg = 6. Answer: 8.
  4. Is 7, 24, 25 a right-triangle triple? Answer: Yes, 49 + 576 = 625.
  5. A 9 m ladder reaches 8 m up a wall. How far is its foot from the wall? Answer: √(81 − 64) = √17 ≈ 4.12 m.
  6. Find the diagonal of a 9 cm by 12 cm rectangle. Answer: 15 cm.
  7. Do sides 4, 5, 6 form a right triangle? Answer: No, 16 + 25 ≠ 36.
  8. A right triangle has legs 9 and 12. Find its hypotenuse. Answer: 15.

Knowledge check

Choose one answer for each question, then check your answers. Explanations appear after checking.

1. Which side is the hypotenuse?
2. What is the hypotenuse when legs are 3 and 4?
3. A right triangle has c = 13 and a = 5. Find b.
4. Which set can be the sides of a right triangle?
5. A rectangle is 5 by 12. What is its diagonal?

Printable worksheet

Use the independent-practice questions above as your worksheet. Show each step, include units where relevant, and check answers only after attempting the questions.

Answer key

  1. Find the hypotenuse when legs are 6 and 8. Answer: √(36 + 64) = 10.
  2. Find the hypotenuse for legs 5 and 12. Answer: 13.
  3. Find the missing leg when c = 10 and one leg = 6. Answer: 8.
  4. Is 7, 24, 25 a right-triangle triple? Answer: Yes, 49 + 576 = 625.
  5. A 9 m ladder reaches 8 m up a wall. How far is its foot from the wall? Answer: √(81 − 64) = √17 ≈ 4.12 m.
  6. Find the diagonal of a 9 cm by 12 cm rectangle. Answer: 15 cm.
  7. Do sides 4, 5, 6 form a right triangle? Answer: No, 16 + 25 ≠ 36.
  8. A right triangle has legs 9 and 12. Find its hypotenuse. Answer: 15.

Study tip: Write down the rule you used beside each answer. This makes errors easier to find and correct.

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