Pythagorean Theorem

📐 Understanding the Theorem

Part 1 of 7 — Understanding the Theorem

In a right triangle with legs $a$ and $b$ and hypotenuse $c$:

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

The hypotenuse is always the longest side, opposite the right angle.

Worked Example

Legs: 3 and 4. Find hypotenuse.

$3^2 + 4^2 = 9 + 16 = 25$ $c=s$ ✅

Concept Check 🎯

Find the Hypotenuse 🧮

1. Legs 3, 4. c = ?

2. Legs 5, 12. c = ?

3. Legs 6, 8. c = ?

Concept Check 🔍

Practice

Challenge Question 📋

📊 Finding Missing Sides

Part 2 of 7 — Finding Missing Sides

To find a leg: $a = sqrt{c^2 - b^2}$

Always identify: which side is the hypotenuse?

Worked Example

Hypotenuse 10, one leg 6. Find the other leg.

✅

🔢 Distance Between Points

Part 3 of 7 — Distance Between Points

The distance formula comes from the Pythagorean theorem:

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

📈 Converse of Pythagorean Theorem

Part 4 of 7 — Converse of Pythagorean Theorem

If $a^2 + b^2 = c^2$, the triangle is a right triangle.

If $$, it's . If , it's .

🧮 3D Applications

Part 5 of 7 — 3D Applications

The Pythagorean theorem extends to 3D:

$d = sqrt{l^2 + w^2 + h^2}$

🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Real-world Pythagorean theorem:

• Ladder against a wall
• Television screen size (diagonal)
• Walking shortest path

Worked Example

Ladder: 10 ft long, base 6 ft from wall. How high does it reach?

$h = sqrt{10^2 - 6^2} = sqrt{100-36} = sqrt{64} = 8$ ft ✅

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• $a^2 + b^2 = c^2$
• Leg: $$