📐 Pythagorean Theorem

Question 1

Why does
a^2 + b^2 = c^2 work for right triangles?

Answer 1

Picture a right triangle with squares built on each side. The area of the big square (on the hypotenuse) is exactly the sum of the areas of the two smaller squares (on the legs). That’s why the formula works.

Question 2

Example: Right triangle with legs 3 and 4. Find the hypotenuse.

Hypotenuse: the longest side of a right triangle, opposite the right angle.

Answer 2

Plug in: 3^2 + 4^2 = 9 + 16 = 25. The hypotenuse = √25 = 5.
So the triangle sides are 3-4-5 (a common “Pythagorean triple”).

Question 3

How does the theorem help find distance between 2 points?

Answer 3

Example: (1,2) and (5,5).
The “run” = 5−1 = 4, the “rise” = 5−2 = 3. That’s a right triangle!
So distance = √(4²+3²) = √25 = 5.