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.

$3^2 + 4^2 = 9 + 16 = 25$ $c = \sqrt{25} = 5$ ✅

Concept Check 🎯

Find the Hypotenuse 🧮

1. Legs 3, 4. c = ?

2. Legs 5, 12. c = ?

3. Legs 6, 8. c = ?

Concept Check 🔍

Practice

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📊 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.

$a = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$ ✅

Concept Check 🎯

Find the Missing Leg 🧮

1. c=13, a=5. b = ?

2. c=15, a=9. b = ?

3. c=17, a=8. b = ?

Concept Check 🔍

Practice

Challenge Question 📋

🔢 Distance Between Points

Part 3 of 7 — Distance Between Points

The distance formula comes from the Pythagorean theorem:

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

The horizontal and vertical differences form the legs of a right triangle.

Worked Example

Distance between (1, 2) and (4, 6).

$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = \sqrt{25} = 5$ ✅

Concept Check 🎯

Find the Distance 🧮

1. (0,0) to (3,4). d = ?

2. (1,2) to (4,6). d = ?

3. (0,0) to (5,12). d = ?

Concept Check 🔍

Practice

Challenge Question 📋

📈 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 $a^2 + b^2 > c^2$, it's acute. If $a^2 + b^2 < c^2$, it's obtuse.

Worked Example

Sides 7, 24, 25. Right triangle?

$7^2 + 24^2 = 49 + 576 = 625 = 25^2$ → Yes, right triangle! ✅

Concept Check 🎯

Check: $a^2 + b^2 = ?$ 🧮

1. a=7, b=24. $a^2 + b^2 = ?$

2. a=5, b=12. $a^2 + b^2 = ?$

3. a=6, b=8. $a^2 + b^2 = ?$

Concept Check 🔍

Practice

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🧮 3D Applications

Part 5 of 7 — 3D Applications

The Pythagorean theorem extends to 3D:

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

Space diagonal of a box: finds the longest line from corner to opposite corner.

Worked Example

Box 3×4×12. Space diagonal?

$d = \sqrt{3^2+4^2+12^2} = \sqrt{9+16+144} = \sqrt{169} = 13$ ✅

Concept Check 🎯

Space Diagonals 🧮

1. Box 3×4×12. Diagonal?

2. Box 1×2×2. Diagonal?

3. Box 2×6×9. Diagonal?

Concept Check 🔍

Practice

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🛠️ 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 ✅

Concept Check 🎯

Word Problems 🧮

1. Ladder 13 ft, base 5 ft from wall. Height?

2. TV screen 16 × 12. Diagonal?

3. Walk 9 blocks east, 12 blocks north. Direct distance?

Concept Check 🔍

Practice

Challenge Question 📋

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Formulas

• $a^2 + b^2 = c^2$
• Leg: $a = \sqrt{c^2 - b^2}$
• Distance: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
• 3D diagonal: $d = \sqrt{l^2+w^2+h^2}$
• Converse: classify triangle by comparing $a^2+b^2$ to $c^2$

Worked Example

Legs 8 and 15. Hypotenuse? $c = \sqrt{64+225} = \sqrt{289} = 17$ ✅

Concept Check 🎯

Review 🧮

1. Legs 8, 15. c = ?

2. c=25, a=7. b = ?

3. Distance: (0,0) to (6,8). d = ?

Concept Check 🔍

Practice

Challenge Question 📋