A right triangle is a triangle with one 90-degree angle. It has three sides: two legs labeled 'a' and 'b', and the hypotenuse labeled 'c'. The hypotenuse is always the longest side and is located opposite to the right angle. Let's look at some examples of right triangles in different orientations. Notice how the hypotenuse is always the side opposite the right angle, regardless of the triangle's position.
The Pythagorean Theorem is one of the most famous mathematical relationships. It states that in any right triangle, the square of the hypotenuse equals the sum of squares of the two legs. The formula is a-squared plus b-squared equals c-squared. Let's see how each part of this formula corresponds to the sides of our triangle. The variable 'a' represents one leg, 'b' represents the other leg, and 'c' represents the hypotenuse. This fundamental relationship works for every right triangle, regardless of its size.
Now let's prove the Pythagorean theorem visually by constructing squares on each side of a right triangle. We'll use a 3-4-5 triangle as our example. First, we draw squares on each side of the triangle. The square on side 'a' has area a-squared, which equals 9. The square on side 'b' has area b-squared, which equals 16. The square on the hypotenuse 'c' has area c-squared, which equals 25. Now watch what happens when we add the areas of the two smaller squares: 9 plus 16 equals 25, which is exactly the area of the largest square. This visual proof demonstrates that a-squared plus b-squared equals c-squared.
Now let's apply the Pythagorean theorem to solve for unknown sides. There are three types of problems. First, finding the hypotenuse when both legs are known. Given a equals 3 and b equals 4, we use c-squared equals a-squared plus b-squared. So c-squared equals 9 plus 16, which equals 25. Taking the square root, c equals 5. Second, finding a leg when the hypotenuse and other leg are known. Given b equals 12 and c equals 13, we rearrange to get a-squared equals c-squared minus b-squared. So a-squared equals 169 minus 144, which equals 25. Therefore a equals 5. Remember, always identify which side is the hypotenuse before solving.
The Pythagorean theorem has countless real-world applications. Let's look at some practical examples. First, finding diagonal distances across rectangular areas. If we have a rectangular field that's 60 meters by 80 meters, we can find the diagonal distance using the theorem. The diagonal equals the square root of 60-squared plus 80-squared, which equals 100 meters. Second, determining safe ladder placement. If we have a 5-meter ladder and place its base 2.4 meters from the wall, we can calculate how high it reaches. The height equals the square root of 5-squared minus 2.4-squared, which is approximately 4.24 meters. These applications show how the Pythagorean theorem helps solve practical problems in construction, navigation, and engineering.