Pythagorean Theorem Explained with Animations

The Pythagorean theorem shows the relationship between the sides of a right triangle.In any right triangle, we label the longest side c, which is called the hypotenuse.The other two sides are labeled a and b.The theorem states that if we create squares on each side of the triangle...The square on side a has an area of a squared, which is nine square units.The square on side b has an area of b squared, which is sixteen square units.And the square on the hypotenuse has an area of c squared, which is twenty-five square units.The Pythagorean theorem states that a squared plus b squared equals c squared.In this case, nine plus sixteen equals twenty-five.Notice how the areas of the squares on the shorter sides...When added together, equal exactly the area of the square on the hypotenuse.The measurements of our triangle are three, four, and five units.This forms a perfect right triangle, demonstrating one of the most famous examples of the Pythagorean theorem.Now let's prove the Pythagorean theorem using areas.We start with a large square with sides of length a plus b.Inside this square, we can arrange four identical right triangles.These triangles form a smaller square in the middle, with sides of length c.The total area of the large square can be written as a plus b, squared.This expands to a squared plus two a b plus b squared.The inner square has an area of c squared.Each triangle has an area of one-half a b.Now, let's rearrange these triangles to show how the areas relate.The total area equals c squared plus the areas of all four triangles.When we expand this, we get a squared plus two a b plus b squared equals c squared plus two a b.The two a b terms cancel out on both sides, leaving us with a squared plus b squared equals c squared.Let's explore how the Pythagorean theorem is used in everyday life.When measuring TV or monitor screen sizes, manufacturers use the diagonal measurement. A TV that's 32 inches wide and 18 inches tall has a diagonal size of 37 inches, calculated using the Pythagorean theorem.On maps, we can find the direct distance between two points using the same principle. The horizontal and vertical distances form the legs of a right triangle.In construction, the three-four-five rule is a practical application that helps create perfect right angles. By measuring 3 units in one direction and 4 units in the perpendicular direction, the diagonal should be exactly 5 units for a perfect right angle.GPS systems use triangulation based on the Pythagorean theorem. By calculating distances from multiple satellites, your device can determine its exact position on Earth.