Pythagorean Theorem Study Notes

The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the three sides of a right-angled triangle. It is named after the ancient Greek mathematician Pythagoras.

📐 What is a Right-Angled Triangle?

A right-angled triangle (also known as a right triangle) is a triangle in which one of the angles is exactly 90 degrees (a right angle).

• The side opposite the right angle is called the hypotenuse. It is always the longest side of the right triangle.
• The other two sides are called legs.

📌 Key Takeaway: Identifying Sides

Imagine a right triangle. The two sides that form the 90-degree angle are the legs. The side that "opens up" to the 90-degree angle is the hypotenuse.

✨ The Pythagorean Theorem Formula

The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

If we label the legs as \(a\) and \(b\), and the hypotenuse as \(c\), the formula is:

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

Where:

• \(a\) and \(b\) are the lengths of the legs.
• \(c\) is the length of the hypotenuse.

💡 Remember: This theorem only applies to right-angled triangles.

✍️ How to Use the Pythagorean Theorem

1. Finding the Hypotenuse (c)

If you know the lengths of the two legs, you can find the length of the hypotenuse.

Example: A right triangle has legs of length 3 units and 4 units. Find the length of the hypotenuse.

• Let \(a = 3\) and \(b = 4\).
• Using the formula: \(a^2 + b^2 = c^2\)
• Substitute the values: \(3^2 + 4^2 = c^2\)
• Calculate the squares: \(9 + 16 = c^2\)
• Add them: \(25 = c^2\)
• Take the square root of both sides: \(\sqrt{25} = \sqrt{c^2}\)
• Result: \(c = 5\) units.

2. Finding a Leg (a or b)

If you know the length of one leg and the hypotenuse, you can find the length of the other leg.

Example: A right triangle has a hypotenuse of length 10 units and one leg of length 6 units. Find the length of the other leg.

• Let \(c = 10\) and \(a = 6\).
• Using the formula: \(a^2 + b^2 = c^2\)
• Substitute the values: \(6^2 + b^2 = 10^2\)
• Calculate the squares: \(36 + b^2 = 100\)
• Subtract 36 from both sides: \(b^2 = 100 - 36\)
• Simplify: \(b^2 = 64\)
• Take the square root of both sides: \(\sqrt{b^2} = \sqrt{64}\)
• Result: \(b = 8\) units.

🔢 Pythagorean Triples

A Pythagorean triple is a set of three positive integers \(a\), \(b\), and \(c\), such that \(a^2 + b^2 = c^2\). These are common side lengths for right triangles.

Here are some common Pythagorean triples:

Leg 1 (a)	Leg 2 (b)	Hypotenuse (c)
3	4	5
5	12	13
8	15	17
7	24	25

🔄 Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem helps us determine if a triangle is a right-angled triangle when we know all three side lengths.

• If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
• In other words, if for a triangle with sides \(a, b, c\) where \(c\) is the longest side, \(a^2 + b^2 = c^2\), then the triangle has a right angle opposite side \(c\).

Example: A triangle has sides of length 7, 9, and 12. Is it a right-angled triangle?

• Identify the longest side: \(c = 12\). The other sides are \(a = 7\) and \(b = 9\).
• Check if \(a^2 + b^2 = c^2\):
• \(7^2 + 9^2 = 12^2\)
• \(49 + 81 = 144\)
• \(130 = 144\)
• Since \(130 \neq 144\), the triangle is not a right-angled triangle.