★ Chapter 7: Trigonometry – Pythagoras’ Theorem ★
Hello mathematicians! Welcome to the fundamental building block of trigonometry and geometry: Pythagoras' theorem. Don't worry if geometry often feels abstract; this theorem is incredibly practical and easy to use once you know the rules. It helps us find missing lengths in one specific type of triangle, making it essential for solving problems in 2D space (and later, in 3D too!).
This chapter will focus purely on knowing and using the formula, including specific applications in coordinate geometry and circles, as required by the IGCSE syllabus.
1. Understanding the Right-Angled Triangle
Pythagoras' theorem is special because it only works for triangles that contain a 90-degree angle. These are called right-angled triangles.
Key Terminology
- Right Angle: The square box symbol (or 90°) indicates this corner.
- Legs (or Shorter Sides): The two sides that meet to form the right angle. We usually call these \(a\) and \(b\).
- Hypotenuse (c): This is the most important side! It is always the longest side of the triangle, and it is always opposite the right angle.
Analogy: Think of the hypotenuse like the main staircase in a building; it's the longest path between the two points, and it faces the main entrance (the right angle).
Quick Review Box: The Golden Rule
The theorem only works if the triangle has a right angle. Always identify the hypotenuse (\(c\)) first!
2. The Formula: Know and Use Pythagoras’ Theorem
The theorem states that for any right-angled triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)).
The Pythagorean Formula
\[a^2 + b^2 = c^2\]
This simple formula allows you to find any side length if you know the other two.
2.1 Case 1: Finding the Hypotenuse (The Longest Side)
If you know the two shorter sides, \(a\) and \(b\), you can find the hypotenuse, \(c\).
Step-by-Step Guide: Finding \(c\)
- Square the length of side \(a\).
- Square the length of side \(b\).
- Add the results from step 1 and 2 together. This gives you \(c^2\).
- Take the square root of the total to find the length of \(c\).
Formula rearranged for \(c\): \[c = \sqrt{a^2 + b^2}\]
Memory Aid: If you are finding the Longest side (Hypotenuse), you Add the squares.
2.2 Case 2: Finding a Shorter Side (a or b)
If you know the hypotenuse (\(c\)) and one shorter side (say, \(a\)), you can find the remaining shorter side (\(b\)).
Since \(a^2 + b^2 = c^2\), we must rearrange the formula to find the missing short side:
\[b^2 = c^2 - a^2\]
Step-by-Step Guide: Finding \(a\) or \(b\)
- Square the hypotenuse, \(c\).
- Square the known shorter side, \(a\).
- Subtract the smaller squared value (from step 2) from the larger squared value (from step 1). This gives you \(b^2\).
- Take the square root of the result to find the length of \(b\).
Formula rearranged for a shorter side: \[a = \sqrt{c^2 - b^2}\]
Memory Aid: If you are finding a Shorter side, you Subtract the squares.
3. Applications of Pythagoras’ Theorem
Pythagoras is not just for abstract triangles! The syllabus requires you to apply it in specific geometric and coordinate contexts.
3.1 Finding the Distance Between Two Points on a Grid
If you have two points, \(A(x_1, y_1)\) and \(B(x_2, y_2)\), you can find the straight-line distance between them by creating a right-angled triangle using the difference in the coordinates.
The horizontal change (\(\Delta x\)) and the vertical change (\(\Delta y\)) become the two shorter sides (\(a\) and \(b\)). The distance, \(d\), is the hypotenuse (\(c\)).
Horizontal distance (\(a\)): \(|x_2 - x_1|\)
Vertical distance (\(b\)): \(|y_2 - y_1|\)
Distance Formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Example:
Find the distance between P(1, 2) and Q(5, 5).
1. Horizontal change: \(\Delta x = 5 - 1 = 4\)
2. Vertical change: \(\Delta y = 5 - 2 = 3\)
3. Use Pythagoras: \(d^2 = 4^2 + 3^2\)
4. \(d^2 = 16 + 9 = 25\)
5. \(d = \sqrt{25} = 5\)
Key Takeaway: The distance formula is just Pythagoras' theorem applied to coordinate geometry.
3.2 Pythagoras in Circle Geometry (Chords)
When dealing with circles, Pythagoras' theorem is often used in combination with geometric properties:
- The radius (\(r\)) is the hypotenuse.
- The line segment drawn from the center of the circle to the chord is perpendicular to the chord (forming the right angle).
- This perpendicular line also bisects (cuts exactly in half) the chord.
This creates a right-angled triangle where the sides are:
- \(a\) = Distance from the centre to the chord.
- \(b\) = Half the length of the chord.
- \(c\) = The radius of the circle.
Example Scenario (Syllabus Check): If a circle has a radius of 10 cm and a chord is 16 cm long, find the distance of the chord from the centre.
- Hypotenuse \(c\) (Radius) = 10 cm
- Shorter side \(b\) (Half-chord) = \(16 \div 2 = 8\) cm
- Missing side \(a\) (Distance from centre) = ?
Calculation: \(a^2 + 8^2 = 10^2\)
\(a^2 + 64 = 100\)
\(a^2 = 100 - 64 = 36\)
\(a = \sqrt{36} = 6\) cm.
Key Takeaway: Drawing the radius to the end of the chord and the perpendicular line from the center creates the perfect right-angled triangle for calculation.
4. Common Mistakes and Pro Tips
❌ Common Mistakes to Avoid:
- Swapping Hypotenuse and Leg: The most common error is using \(a^2 + b^2 = c^2\) but plugging the hypotenuse value into \(a\) or \(b\). Always confirm \(c\) is the longest side opposite the 90° angle.
- Forgetting the Square Root: You calculate \(c^2\) or \(a^2\), but forget the final step of taking the square root to get the actual length.
- Mixing Addition and Subtraction: Remember: Addition is for finding the hypotenuse; Subtraction is for finding a shorter side.
✅ Study Tips: Pythagorean Triples
A Pythagorean Triple is a set of three integers that perfectly satisfy \(a^2 + b^2 = c^2\). Knowing these can save you time in non-calculator papers or help you quickly check your work.
- The Famous One: (3, 4, 5). (\(3^2 + 4^2 = 9 + 16 = 25 = 5^2\))
- Other Common Ones: (5, 12, 13) and (8, 15, 17).
Did you know? Any multiple of a triple is also a triple! For example, (6, 8, 10) is just 2 times (3, 4, 5).
Summary and Next Steps
Pythagoras' theorem is your essential tool for finding missing side lengths in any right-angled triangle. It provides the foundation upon which the rest of trigonometry is built.
Key Takeaway
Always start by identifying the hypotenuse (\(c\)). Use addition to find \(c\) (\(c^2 = a^2 + b^2\)), and use subtraction to find a shorter leg (\(a^2 = c^2 - b^2\)). Ensure your final answer is reasonable: the hypotenuse must be the longest side.