What is the Pythagorean Theorem?

The Pythagorean 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. In simpler terms, $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the two shorter sides (legs) and 'c' is the length of the longest side (hypotenuse).

📐 Understanding Right-Angled Triangles

• The theorem applies only to right-angled triangles, which have one angle that measures exactly 90 degrees.
• The hypotenuse is always the side opposite the right angle and is always the longest side.
• The other two sides are often referred to as the legs of the right triangle.

🚀 How it works

1. Identify the right angle and the hypotenuse (side 'c').
2. Measure the lengths of the two legs (sides 'a' and 'b').
3. Square the lengths of each leg (calculate $a^2$ and $b^2$).
4. Add the squares of the legs ($a^2 + b^2$).
5. The result is equal to the square of the hypotenuse ($c^2$). To find the actual length of the hypotenuse, take the square root of this sum (calculate $c = \sqrt{a^2 + b^2}$).

🍎 Real-World Example

Imagine you're leaning a 13-foot ladder against a wall. The base of the ladder is 5 feet away from the wall. How high up the wall does the ladder reach?

• The ladder is the hypotenuse (c = 13 feet).
• The distance from the wall is one leg (a = 5 feet).
• We need to find the height up the wall (side 'b').

Using the theorem: $a^2 + b^2 = c^2$, so $5^2 + b^2 = 13^2$.

This simplifies to $25 + b^2 = 169$.

Subtracting 25 from both sides: $b^2 = 144$.

Taking the square root of both sides: $b = \sqrt{144} = 12$ feet.

Therefore, the ladder reaches 12 feet up the wall.

💡 Pro Tip: The Pythagorean Theorem is not just limited to geometry problems. Engineers, architects, and even navigators use it daily for calculations involving distances and angles!

✨ Common Uses

• Construction: Ensuring buildings are square and stable.
• Navigation: Calculating distances and routes.
• Engineering: Designing bridges and structures.
• Mathematics: Solving geometric problems.