How to Solve: Pythagoras Theorem

How to Solve: Pythagoras Theorem

Introduction

"Ever wondered how builders check if a corner is perfectly square, or how GPS calculates the shortest distance between two points? Mastering Pythagoras’ Theorem unlocks these real-world problems—and guarantees you full marks on exam questions about right-angled triangles!

What You Need To Know First

Before diving into Pythagoras’ Theorem, ensure you understand: 1. Right-angled triangles – A triangle with one 90° angle (the "right angle"). 2. Hypotenuse – The side opposite the right angle (the longest side). 3. Square numbers and square roots – How to calculate ( a^2 ) and ( \sqrt{a} ).

Key Vocabulary

Formulas To Know

1. Pythagoras’ Theorem

Formula: [ a^2 + b^2 = c^2 ]

Variables: - ( a ) and ( b ) = lengths of the two legs (shorter sides). - ( c ) = length of the hypotenuse (longest side, opposite the right angle).

Memorise this? ✅ YES – MUST MEMORISE.

2. Rearranged Formulas (for finding a leg)

If you need to find a leg instead of the hypotenuse: [ a^2 = c^2 - b^2 ] [ b^2 = c^2 - a^2 ]

Memorise this? ❌ Not necessary—just rearrange the main formula.

Step-by-Step Method

Step 1: Identify the right-angled triangle

• Look for the right-angle symbol (□) in the diagram.
• If no diagram is given, check if the question mentions a right-angled triangle or perpendicular sides.

Step 2: Label the sides

• Hypotenuse (c): The side opposite the right angle (longest side).
• Legs (a and b): The other two sides (order doesn’t matter).

Step 3: Write down the formula

[ a^2 + b^2 = c^2 ]

Step 4: Substitute the known values

• Plug in the numbers for ( a ), ( b ), or ( c ).
• If a side is missing, leave it as a variable (e.g., ( x )).

Step 5: Solve for the unknown

• If finding the hypotenuse: Add ( a^2 + b^2 ), then take the square root.
• If finding a leg: Subtract ( c^2 - b^2 ) (or ( c^2 - a^2 )), then take the square root.

Step 6: Check your answer

• The hypotenuse must be the longest side.
• If your answer doesn’t make sense (e.g., negative length), recheck your steps.

Step 7: Write the final answer with units

• Always include units (e.g., cm, m, km).

Worked Example Using the Steps

Question: A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.

Step 1: Identify the right-angled triangle. - The question states it’s a right-angled triangle.

Step 2: Label the sides. - Legs: ( a = 6 ) cm, ( b = 8 ) cm. - Hypotenuse: ( c = ? ).

Step 3: Write the formula. [ a^2 + b^2 = c^2 ]

Step 4: Substitute the known values. [ 6^2 + 8^2 = c^2 ] [ 36 + 64 = c^2 ]

Step 5: Solve for ( c ). [ 100 = c^2 ] [ c = \sqrt{100} ] [ c = 10 ]

Step 6: Check the answer. - 10 cm is longer than 6 cm and 8 cm (correct for hypotenuse).

Step 7: Write the final answer. Answer: The hypotenuse is 10 cm.

Worked Examples

Example 1 – Basic (Finding the Hypotenuse)

Question: A right-angled triangle has legs of 5 m and 12 m. Find the hypotenuse.

Working: 1. Label sides: ( a = 5 ) m, ( b = 12 ) m, ( c = ? ). 2. Formula: ( a^2 + b^2 = c^2 ). 3. Substitute: ( 5^2 + 12^2 = c^2 ). 4. Calculate: ( 25 + 144 = c^2 ). 5. ( 169 = c^2 ). 6. ( c = \sqrt{169} = 13 ).

Answer: The hypotenuse is 13 m.

What we did and why: - We used the theorem directly to find the hypotenuse. - Recognised that ( 5, 12, 13 ) is a Pythagorean triple (saves time in exams!).

Example 2 – Medium (Finding a Leg)

Question: A right-angled triangle has a hypotenuse of 15 cm and one leg of 9 cm. Find the other leg.

Working: 1. Label sides: ( a = 9 ) cm, ( c = 15 ) cm, ( b = ? ). 2. Formula: ( a^2 + b^2 = c^2 ). 3. Rearrange: ( b^2 = c^2 - a^2 ). 4. Substitute: ( b^2 = 15^2 - 9^2 ). 5. Calculate: ( b^2 = 225 - 81 = 144 ). 6. ( b = \sqrt{144} = 12 ).

Answer: The missing leg is 12 cm.

What we did and why: - We rearranged the formula to solve for a leg instead of the hypotenuse. - Always subtract the known leg’s square from the hypotenuse’s square.

Example 3 – Exam Style (Disguised Question)

Question: A ladder leans against a wall. The base of the ladder is 3 m from the wall, and the top reaches 4 m up the wall. How long is the ladder?

Working: 1. Draw a diagram:
- Wall = vertical side (4 m).
- Ground = horizontal side (3 m).
- Ladder = hypotenuse. 2. Label sides: ( a = 3 ) m, ( b = 4 ) m, ( c = ? ). 3. Formula: ( a^2 + b^2 = c^2 ). 4. Substitute: ( 3^2 + 4^2 = c^2 ). 5. Calculate: ( 9 + 16 = c^2 ). 6. ( 25 = c^2 ). 7. ( c = \sqrt{25} = 5 ).

Answer: The ladder is 5 m long.

What we did and why: - We translated a word problem into a right-angled triangle. - Recognised that ( 3, 4, 5 ) is a common Pythagorean triple (quick check!).

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—tonight or right before your exam. Here’s the one-minute Pythagoras cheat sheet:

1. Spot the right angle—if it’s not there, you can’t use this!
2. Label the sides—hypotenuse is the longest, opposite the right angle. The other two are legs.
3. Write the formula: ( a^2 + b^2 = c^2 ). Memorise it.
4. Plug in the numbers—if you’re missing a leg, rearrange to ( a^2 = c^2 - b^2 ).
5. Square, add (or subtract), then square root—don’t skip steps!
6. Check your answer—the hypotenuse must be the longest side.

Pro tip: If the numbers are 3, 4, 5 or 5, 12, 13, you’ve got a Pythagorean triple—no calculator needed!

Exam day? Draw a quick diagram, label everything, and show all working. Examiners love that. Now go ace it!