Anjeli writes the equation (a + b)² = c² + 4(1/2 ab) to begin a proof of the Pythagorean theorem.
Use the drop-down menus to explain why this is a true equation.
(a + b)² finds the area of the outer square by squaring its side length.
+4(1/2 ab) finds the area of the outer square by adding the areas of the inner square and 4 triangles.
These expressions are equal because they both give the area of the outer square.
The equation (a + b)² = c² + 4(1/2 ab) is true because both sides represent the area of the same geometric figure, which is a square formed by the sides of lengths a and b.
1. The left side, (a + b)², calculates the area of the outer square, where the side length is the sum of a and b. When you square this sum, you find the total area of the square.
2. The right side, c² + 4(1/2 ab), shows the area of the outer square as well, where c² represents the area of the inner square (which has a side length of c) and the term 4(1/2 ab) accounts for the areas of the four right triangles formed around the inner square, each with an area of (1/2 ab).
Since both sides represent the total area of the same outer square, the equation is true.