How do you find the hypotenuse of a right triangle?

To find the hypotenuse (c), use the formula c = √(a² + b²). Square both legs, add them together, then take the square root. For example, if a = 3 and b = 4, then c = √(3² + 4²) = √(9 + 16) = √25 = 5.

How do you find a missing leg in a right triangle?

To find a missing leg when you know the hypotenuse and the other leg, use: a = √(c² - b²) or b = √(c² - a²). For example, if c = 5 and b = 4, then a = √(5² - 4²) = √(25 - 16) = √9 = 3.

What is a 3-4-5 triangle?

A 3-4-5 triangle is the most famous Pythagorean triple - a right triangle with sides of length 3, 4, and 5. It demonstrates the theorem: 3² + 4² = 9 + 16 = 25 = 5². Other common Pythagorean triples include 5-12-13 and 8-15-17.

How do you verify if a triangle is a right triangle?

To verify if a triangle is a right triangle, check if the Pythagorean theorem holds: a² + b² = c² (where c is the longest side). If the equation is true, it's a right triangle. If not, it's either acute (a² + b² > c²) or obtuse (a² + b² < c²).

Who discovered the Pythagorean theorem?

While named after the Greek mathematician Pythagoras (c. 570-495 BCE), the theorem was known to earlier civilizations including the Babylonians and Indians. Pythagoras is credited with providing the first formal proof of the theorem in ancient Greece.

Pythagorean Theorem Calculator - Find Missing Side of Right Triangle | AI Calculator

The Pythagorean theorem is one of the most fundamental and widely used principles in mathematics. Named after the ancient Greek mathematician Pythagoras, this theorem describes the relationship between the sides of a right triangle. It states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of squares of the other two sides.

The Pythagorean Theorem Formula

a² + b² = c²

a and b = lengths of the two legs (sides forming the right angle)

c = length of the hypotenuse (longest side, opposite the right angle)

Common Applications

🔍 Finding the Hypotenuse

Formula: c = √(a² + b²)

When to use: You know both legs

Example: a = 3, b = 4 → c = √(9 + 16) = 5

📏 Finding a Leg

Formula: a = √(c² - b²)

When to use: You know hypotenuse and one leg

Example: c = 5, b = 4 → a = √(25 - 16) = 3

✓ Verification

Check: Does a² + b² = c²?

When to use: Verify if triangle is right

Example: 3² + 4² = 9 + 16 = 25 = 5² ✓

Famous Pythagorean Triples

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. These combinations are particularly useful in construction, navigation, and various practical applications.

Common Pythagorean Triples

Basic Triples

• 3-4-5: 3² + 4² = 9 + 16 = 25 = 5²
• 5-12-13: 5² + 12² = 25 + 144 = 169 = 13²
• 8-15-17: 8² + 15² = 64 + 225 = 289 = 17²
• 7-24-25: 7² + 24² = 49 + 576 = 625 = 25²

Scaled Multiples

• 6-8-10: 2 × (3-4-5)
• 9-12-15: 3 × (3-4-5)
• 10-24-26: 2 × (5-12-13)
• 15-20-25: 5 × (3-4-5)

Real-World Applications

🏗️ Construction & Engineering

• Ensuring structures are square and level
• Calculating roof rafter lengths
• Determining diagonal bracing requirements
• Foundation layout and corner verification
• Staircase design and handrail placement

🧭 Navigation & Surveying

• GPS distance calculations
• Property boundary measurements
• Ship and aircraft navigation
• Triangulation in mapping
• Shortest path calculations

Step-by-Step Problem Solving

Example 1: Finding the Hypotenuse

Problem: A right triangle has legs of 6 and 8 units. Find the hypotenuse.

Solution:

1. Use the formula: c = √(a² + b²)

2. Substitute: c = √(6² + 8²) = √(36 + 64) = √100

3. Calculate: c = 10 units

Verification: 6² + 8² = 36 + 64 = 100 = 10² ✓

Example 2: Finding a Missing Leg

Problem: A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

Solution:

1. Use the formula: b = √(c² - a²)

2. Substitute: b = √(13² - 5²) = √(169 - 25) = √144

3. Calculate: b = 12 units

Verification: 5² + 12² = 25 + 144 = 169 = 13² ✓

Historical Context

While the theorem bears Pythagoras's name (c. 570-495 BCE), evidence suggests that the relationship was known to earlier civilizations. Babylonian tablets from around 1800 BCE show knowledge of Pythagorean triples, and ancient Indian texts also reference the theorem. Pythagoras is credited with providing the first formal mathematical proof in ancient Greece.

💡 Tips for Using the Pythagorean Theorem

• Always identify the hypotenuse first - it's the longest side opposite the right angle
• Remember that the theorem only applies to right triangles (90° angle)
• When finding a leg, ensure the hypotenuse is longer than the known leg
• Use Pythagorean triples for quick mental calculations
• Double-check your work by verifying a² + b² = c²
• In practical applications, consider measurement precision and rounding

Beyond Basic Applications

The Pythagorean theorem extends beyond simple triangle calculations. It forms the foundation for distance formulas in coordinate geometry, appears in vector mathematics, and is essential in trigonometry. In three dimensions, it generalizes to the distance formula: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²].