Summary The Pythagorean Theorem
**Pythagorean Theorem**
The Pythagorean theorem is a fundamental relation in Euclidean geometry that 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. **Formula:**
```
a² + b² = c²
```
where:
* `a` and `b` are the lengths of the two shorter sides (legs) of the right triangle
* `c` is the length of the hypotenuse
**Applications:**
The Pythagorean theorem has numerous applications in various fields, including:
* **Geometry:** Finding the lengths of sides and angles in right triangles
* **Trigonometry:** Deriving trigonometric identities and solving trigonometric equations
* **Physics:** Calculating distances and velocities in projectile motion and circular motion
* **Engineering:** Designing and analyzing structures
**Proof:**
There are several different proofs of the Pythagorean theorem, one of which is the following:
* Divide the right triangle into two smaller right triangles by drawing an altitude from the right
angle to the hypotenuse.
* Use the area formula for triangles (`A = ½ bh`) to find the areas of the three triangles.
* Set the area of the original triangle equal to the sum of the areas of the two smaller triangles.
* Simplify the resulting equation to obtain the Pythagorean theorem.
**Converse:**
The converse of the Pythagorean theorem also holds true: If the square of the length of one
side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then
the triangle is a right triangle.
