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: Analyze the shaded triangles and white square

There are 4 right - angled triangles, each with area $\frac{1}{2}ab$, and a white square with area $c^{2}$. So the area of the figure as the sum of the area of the shaded triangles and the white square is $4\times\frac{1}{2}ab + c^{2}=2ab + c^{2}$.

Step2: Analyze the area using the side length of the big square

The side length of the big square is $(a + b)$, so its area is $(a + b)^{2}=a^{2}+2ab + b^{2}$.

Step3: Derive the Pythagorean theorem

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

For the first "Choose...": 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.
For the second "Choose...": The equivalent expression $(a + b)^{2}$ (or $a^{2}+2ab + b^{2}$) uses the length of the figure to represent the area.
For the third "Choose...": Setting two of these area expressions equal to each other and subtracting $2ab$ from both sides of the equation results in the Pythagorean Theorem, $a^{2}+b^{2}=c^{2}$.

Answer:

First "Choose...": $2ab + c^{2}$
Second "Choose...": $(a + b)^{2}$ (or $a^{2}+2ab + b^{2}$)
Third "Choose...": $2ab$