Mario writes the equation (x + y)² = z² + 4(½xy) to begin a proof of the Pythagorean theorem. Use the drop-down menus to explain why this is a true equation.
(x + y)² finds the area of the outer square by squaring its side length.
+ 4(½xy) finds the area of the outer square by adding the area of the inner square and the four triangles.
These expressions are equal because they both give the area of the outer square.
The equation ((x + y)^2 = z^2 + 4(½xy)) represents the relationship between the areas of shapes involved in the proof of the Pythagorean theorem.
1. (x + y)² finds the area of the outer square by squaring its side length. This side length is the sum of the legs of a right triangle (let’s say (x) and (y) are the lengths of the legs).
2. 4(½xy) finds the area of the outer square by adding the area of the inner square (z²) (where (z) is the hypotenuse) and the areas of the four right triangles formed at the corners, each with base (x/2) and height (y/2) (hence (½xy) for each triangle).
These expressions are equal because they both calculate the total area of the same outer square, thereby confirming the relationship described in the Pythagorean theorem. This reasoning helps show that the sum of the areas of the squares on the legs is equal to the area of the square on the hypotenuse.
Do you have any more questions on this topic? Feel free to ask!