Question

the figure below can be used to prove the pythagorean theorem. use the drop - down menus to complete the proof. click the buttons to choose an answer from each menu. the expression choose... represents the area of the figure as the sum of the area of the shaded triangles and the area of the white square. the equivalent expressions choose... use the length of the figure to represent the area. setting two of these area expressions equal to each other and subtracting terms both sides of the equation results in the pythagorean theorem, a² + b² = c².

Explanation:

Step1: Calculate area of shaded triangles and white - square

The area of each right - triangle is $\frac{1}{2}ab$. There are 4 such triangles, so the total area of the triangles is $4\times\frac{1}{2}ab = 2ab$. The area of the white square is $c^{2}$. So the area of the whole figure as the sum of the area of the shaded triangles and the area of the white square is $2ab + c^{2}$.

Step2: Calculate area of the whole figure using side - length

The side - length of the large square is $a + b$. So the area of the large square is $(a + b)^{2}=a^{2}+2ab + b^{2}$ using the formula $(x + y)^{2}=x^{2}+2xy + y^{2}$ where $x=a$ and $y = b$.

Step3: Derive the Pythagorean theorem

Set the two area expressions equal: $2ab + c^{2}=a^{2}+2ab + b^{2}$. Subtract $2ab$ from both sides of the equation. We get $c^{2}=a^{2}+b^{2}$.

Answer:

The expression $2ab + c^{2}$ represents the area of the figure as the sum of the area of the shaded triangles and the area of the white square. The equivalent expression $a^{2}+2ab + b^{2}$ uses the length of the figure to represent the area. Setting two of these area expressions equal to each other and subtracting $2ab$ from both sides of the equation results in the Pythagorean theorem $c^{2}=a^{2}+b^{2}$.