Pythagoras’ Theorem

Key Concepts

• Pythagoras’ theorem applies only to a right-angled triangle.

  • A right-angled triangle is a triangle that has one angle of 90°.
  • The theorem states:
  $$a^2 + b^2 = c^2$$

• In the formula:

  • $a$ and $b$ are the lengths of the two shorter sides that form the right angle.
  • $c$ is the length of the hypotenuse.
  • The hypotenuse is always:
    • the longest side of a right-angled triangle
    • the side opposite the 90° angle

• What the formula means:

  • If you square the lengths of the two shorter sides and add them together, you get the square of the hypotenuse.

  • Example:

    $$3^2 + 4^2 = 5^2$$
    $$9 + 16 = 25$$

• Finding the hypotenuse:

  • If the two shorter sides are known, use:

    $$c = \sqrt{a^2 + b^2}$$

  • Steps:

    1. Square each shorter side.
    2. Add the squares.
    3. Take the square root.

• Finding a shorter side:

  • If the hypotenuse and one shorter side are known, rearrange the theorem:

    $$a^2 = c^2 - b^2$$

    or

    $$b^2 = c^2 - a^2$$

  • Then take the square root.

  • Steps:

    1. Square the hypotenuse.
    2. Square the known shorter side.
    3. Subtract.
    4. Take the square root.

• Pythagorean triples are sets of 3 whole numbers that satisfy Pythagoras’ theorem exactly.

  • Common examples:
    • $3, 4, 5$
    • $5, 12, 13$
    • $8, 15, 17$
    • $7, 24, 25$
  • These are useful because they allow quick answers without using a calculator much.

• Applications in real-world contexts:

  • Pythagoras’ theorem is used to find distances that cannot be measured directly.
  • Examples include:
    • the length of a ladder leaning against a wall
    • the diagonal of a rectangle
    • the shortest straight-line distance between two points
    • the height of an object using a sloping side
    • distances on maps or floor plans

• Converse of Pythagoras’ theorem:

  • The converse is used to check whether a triangle is right-angled.

  • If the side lengths of a triangle satisfy:

    $$a^2 + b^2 = c^2$$

    where $c$ is the longest side, then the triangle is a right-angled triangle.

  • If the equation is not true, the triangle is not right-angled.

• Important note about units:

  • All side lengths must be in the same unit before using the formula.
  • Example: convert cm to m, or m to cm, before calculating.

• Important note about square roots:

  • A length cannot be negative.
  • When finding a side length, use the positive square root only.

Important Definitions

• Pythagoras’ theorem: the rule that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, $\,a^2+b^2=c^2$.

• Right-angled triangle: a triangle with one angle equal to 90°.

• Hypotenuse: the longest side of a right-angled triangle, opposite the right angle.

• Shorter sides: the two sides that meet to form the right angle in a right-angled triangle.

• Square of a number: the result of multiplying a number by itself, for example $6^2 = 36$.

• Square root: a number that, when multiplied by itself, gives the original number, for example $\sqrt{49}=7$.

• Pythagorean triple: a set of three whole numbers that satisfy Pythagoras’ theorem exactly.

• Converse of Pythagoras’ theorem: the rule used to test whether a triangle is right-angled by checking whether the side lengths satisfy $\,a^2+b^2=c^2$.

• Diagonal: a straight line joining two opposite corners of a shape such as a rectangle or square.

• Perpendicular: meeting at an angle of 90°.

Worked Examples

Example 1: Finding the hypotenuse

A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.

Step 1: Write the formula

Step 2: Substitute the values

Step 3: Square the numbers

Step 4: Add

Step 5: Take the square root

Answer: The hypotenuse is 10 cm.

Example 2: Finding a shorter side

A right-angled triangle has hypotenuse 13 m and one shorter side 5 m. Find the other shorter side.

Step 1: Write the formula

Let the unknown side be $a$.

Step 2: Square the known numbers

Step 3: Rearrange

Step 4: Take the square root

Answer: The other shorter side is 12 m.

Example 3: Using the converse of Pythagoras’ theorem

Check whether a triangle with side lengths 9 cm, 12 cm, and 15 cm is right-angled.

Step 1: Identify the longest side

• The longest side is 15 cm.
• So let $c = 15$, $a = 9$, $b = 12$.

Step 2: Use the converse Check whether:

Step 3: Square the side lengths

Step 4: Conclude Since both sides are equal, the triangle satisfies Pythagoras’ theorem.

Answer: Yes, the triangle is right-angled.

Common Mistakes to Avoid

• Using Pythagoras’ theorem on a triangle that is not right-angled.
• Choosing the wrong side as the hypotenuse.
  • Remember: the hypotenuse is always opposite the 90° angle and is the longest side.
• Forgetting to square the numbers.
  • Example: writing $3 + 4 = 5$ instead of $3^2 + 4^2 = 5^2$.
• Forgetting to take the square root at the end when finding a side.
  • Example: stopping at $c^2 = 81$ and saying the answer is 81 instead of 9.
• Mixing up the formula when finding a shorter side.

  • Correct method:

    $$\text{shorter side}^2 = \text{hypotenuse}^2 - \text{other shorter side}^2$$
• Subtracting in the wrong order.

  • Always do:

    $$c^2 - \text{shorter side}^2$$

  • Not the other way round.

• Forgetting to write units in the final answer.
• Using different units without converting first.
  • Example: one side in cm and another in m.
• For the converse, not using the longest side as $c$.
• Rounding too early during working, which may cause small errors.

Exam Tips

• First check whether the triangle is right-angled before using Pythagoras’ theorem.

• In your working, clearly state:

  • “Using Pythagoras’ theorem”
  • or “Using the converse of Pythagoras’ theorem”

• Label the sides carefully:

  • $c$ = hypotenuse
  • $a$ and $b$ = shorter sides

• Show substitution clearly:

  $$a^2 + b^2 = c^2$$
  $$5^2 + 12^2 = c^2$$

• If checking whether a triangle is right-angled, write a full conclusion such as:

  • “Since $a^2 + b^2 = c^2$, the triangle is right-angled.”
  • “Since $a^2 + b^2 \ne c^2$, the triangle is not right-angled.”

• For word problems:

  • Identify the horizontal, vertical, and sloping sides.
  • Draw a neat diagram before calculating.

• Write the final answer with:

  • correct units
  • correct degree of accuracy if required

• If the answer is a decimal, keep enough figures during working and round only at the end.

• Memorise common Pythagorean triples:

  • $3,4,5$
  • $5,12,13$
  • $8,15,17$
  • $7,24,25$

Quick Summary

• Pythagoras’ theorem applies only to a right-angled triangle.

• The formula is:

  $$a^2 + b^2 = c^2$$

• The hypotenuse is the side opposite the 90° angle and is the longest side.

• To find the hypotenuse:

  $$c = \sqrt{a^2+b^2}$$

• To find a shorter side:

  $$a = \sqrt{c^2-b^2}$$

  or

  $$b = \sqrt{c^2-a^2}$$

• Common Pythagorean triples include:

  • $3,4,5$
  • $5,12,13$
  • $8,15,17$
  • $7,24,25$

• Real-life applications include ladders, diagonals, map distances, and heights.

• The converse of Pythagoras’ theorem is used to test whether a triangle is right-angled.

• When using the converse, always treat the longest side as $c$.

• Use the same units for all sides before calculating.

• A side length must be positive, so use the positive square root only.

• Always include clear working, units, and a proper final statement in exam answers.