Question

click the arrows 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 choose... from both sides of the equation results in the pythagorean theorem, $a^{2}+b^{2}=c^{2}$.

Explanation:

Step1: Calculate area of shaded triangles and white - square

The two shaded right - triangles are congruent. Area of one right - triangle is $\frac{1}{2}ab$, so area of two is $ab$. Area of white square is $c^{2}$. So the sum of their areas is $ab + c^{2}$.

Step2: Calculate area of large rectangle

The large rectangle has length $a + b$ and width $a + b$, so its area is $(a + b)^{2}=a^{2}+2ab + b^{2}$.

Step3: Set up equation and simplify

Set $ab + c^{2}=a^{2}+2ab + b^{2}$. Subtract $2ab$ from both sides: $c^{2}=a^{2}+b^{2}$.

Answer:

The expression $ab + c^{2}$ 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 $a^{2}+2ab + b^{2}$ (from $(a + b)^{2}$) 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, $a^{2}+b^{2}=c^{2}$.