International Mathematics (0607) | Pythagoras’ theorem

Hello IGCSE Mathematicians! Navigating Pythagoras’ Theorem (0607)

Welcome to the foundational chapter of Trigonometry! Before we jump into sine, cosine, and tangent, we need a solid understanding of the most famous theorem in geometry: Pythagoras’ Theorem.

Why is this important? Pythagoras' Theorem is your mathematical GPS for finding unknown lengths in right-angled situations. Whether you are building a house, designing a bridge, or calculating the shortest distance across a coordinate plane, this rule is essential!

Don't worry if geometry seems tough—we will break this concept down step-by-step so you can master it for your exams.

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1. The Right-Angled Triangle and the Magic Formula

1.1 Identifying the Right Triangle

Pythagoras’ Theorem only works for a very specific type of triangle: the right-angled triangle.

• A right-angled triangle is any triangle that contains one angle measuring exactly 90 degrees.
• This 90° angle is usually marked with a small square box.

1.2 Key Vocabulary: The Sides

In a right-angled triangle, the sides have special names:

1. The Hypotenuse (c):

• This is the longest side of the triangle.
• It is always the side located directly opposite the 90° angle.
• Analogy: Imagine crossing a park. The hypotenuse is the diagonal path—it’s always longer than walking along the two sides (legs)!

2. The Legs (a and b):

• These are the two shorter sides that meet to form the 90° angle.
• We often call them 'a' and 'b'.

1.3 The Formula

The theorem states that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (the legs).

\(a^2 + b^2 = c^2\)

Memory Aid: Always remember that c must stand alone because it is the biggest, most important side (the hypotenuse).

Key Takeaway: Pythagoras' theorem \(a^2 + b^2 = c^2\) is exclusively for right-angled triangles, where \(c\) is the longest side (the hypotenuse).

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2. Using the Theorem: Finding the Hypotenuse

If you know the lengths of the two legs (\(a\) and \(b\)), you can easily find the length of the hypotenuse (\(c\)).

Step-by-Step Process (Finding c)

1. Square the length of side \(a\).
2. Square the length of side \(b\).
3. Add the two squared results together (this gives you \(c^2\)).
4. Take the Square Root of the total to find the length of \(c\).

Example 1: Calculating the length of a ramp

A builder needs to know the length of a diagonal ramp (c). The horizontal length (a) is 3 m and the vertical height (b) is 4 m.

1. Set up the equation: \(a^2 + b^2 = c^2\)
2. Substitute values: \(3^2 + 4^2 = c^2\)
3. Calculate squares: \(9 + 16 = c^2\)
4. Add: \(25 = c^2\)
5. Find the square root: \(c = \sqrt{25}\)
6. Result: \(c = 5\) m

Did you know? The lengths 3, 4, and 5 form a special set of numbers called a Pythagorean Triple. Any multiples of these (like 6, 8, 10 or 30, 40, 50) are also perfect right-angled triangles!

Key Takeaway: To find the hypotenuse, you add the squares of the two shorter sides before taking the final square root.

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3. Using the Theorem: Finding a Shorter Side

What if you know the hypotenuse (\(c\)) and one of the legs (\(a\)), and you need to find the missing leg (\(b\))?

We need to rearrange the original formula:

Start with: \(a^2 + b^2 = c^2\)

Rearrange to find \(b^2\): \(b^2 = c^2 - a^2\) (Hypotenuse squared minus known leg squared)

Step-by-Step Process (Finding a or b)

1. Square the hypotenuse (\(c\)).
2. Square the known leg (\(a\)).
3. Subtract the smaller squared number from the larger squared number (this gives you the missing side squared).
4. Take the Square Root of the result.

Example 2: Finding a height

A 10 m ladder (hypotenuse, c) leans against a wall. The base of the ladder is 6 m away from the wall (leg, a). How high up the wall (leg, b) does the ladder reach?

1. Set up the equation: \(a^2 + b^2 = c^2\)
2. Substitute values: \(6^2 + b^2 = 10^2\)
3. Isolate \(b^2\): \(b^2 = 10^2 - 6^2\)
4. Calculate squares: \(b^2 = 100 - 36\)
5. Subtract: \(b^2 = 64\)
6. Find the square root: \(b = \sqrt{64}\)
7. Result: \(b = 8\) m

Key Takeaway: To find a shorter side, you must subtract the square of the known leg from the square of the hypotenuse.

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4. Pythagoras in Practice: Advanced Applications

The syllabus requires you to apply Pythagoras’ theorem in coordinate geometry and circle geometry.

4.1 Application A: Distance Between Two Points on a Grid

If you are given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the distance between them is simply the hypotenuse of a right-angled triangle formed by the change in \(x\) and the change in \(y\).

The horizontal distance (leg \(a\)) is the difference in the x-coordinates: \(\Delta x = |x_2 - x_1|\)

The vertical distance (leg \(b\)) is the difference in the y-coordinates: \(\Delta y = |y_2 - y_1|\)

Using \(a^2 + b^2 = c^2\), the distance \(d\) is:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Example 3: Calculating distance on a grid

Find the distance between Point A (1, 2) and Point B (5, 9).

1. Find the change in x: \(\Delta x = 5 - 1 = 4\)
2. Find the change in y: \(\Delta y = 9 - 2 = 7\)
3. Apply Pythagoras: \(d^2 = 4^2 + 7^2\)
4. Calculate: \(d^2 = 16 + 49 = 65\)
5. Find the distance: \(d = \sqrt{65}\)
6. Result (to 3 s.f.): \(d \approx 8.06\) units.

4.2 Application B: Lengths of Chords and Distances from the Centre in Circles

Pythagoras is frequently used when dealing with circles, particularly when finding the length of a chord or its distance from the centre.

Remember this essential geometry rule:

Rule: A line segment drawn from the centre of a circle perpendicular (at 90°) to a chord bisects (cuts in half) the chord.

This rule immediately creates a right-angled triangle where:

• The Hypotenuse is always the radius (\(r\)) of the circle.
• One Leg is the distance from the centre to the chord (\(d\)).
• The other Leg is half the length of the chord (\(L/2\)).

\(d^2 + (L/2)^2 = r^2\)

Example 4: Chord distance

A circle has a radius of 5 cm. A chord within the circle is 8 cm long. Find the distance from the centre to the chord.

1. Identify the sides: Radius (\(r\)) = 5 cm (Hypotenuse).
2. Half the chord length (\(L/2\)) = \(8/2 = 4\) cm (Leg).
3. Distance from center (\(d\)) = unknown (Leg).
4. Apply Pythagoras (finding a leg): \(d^2 = 5^2 - 4^2\)
5. Calculate: \(d^2 = 25 - 16 = 9\)
6. Result: \(d = \sqrt{9} = 3\) cm.

Key Takeaway: When solving distance problems (on a grid) or circle problems (involving chords), use the surrounding information (coordinate differences or the radius/half-chord) to create a right-angled triangle and apply \(a^2 + b^2 = c^2\).

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5. Final Review and Study Tips

You’ve got this! Pythagoras’ theorem is a powerful tool, but it requires practice. Here are the main points to solidify your understanding:

Keep practicing these different scenarios—especially the coordinate geometry and circle problems—and you will master Pythagoras’ Theorem in no time!