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 figure's components

The figure has 4 right - angled triangles (shaded) and 1 white square. The area of a right - angled triangle with legs \(a\) and \(b\) is \(\frac{1}{2}ab\). There are 4 such triangles, so the area of the shaded triangles is \(4\times\frac{1}{2}ab = 2ab\). The area of the white square with side \(c\) is \(c^{2}\). So the area of the figure as the sum of the shaded triangles and the white square is \(c^{2}+2ab\).

Step2: Analyze the area using the length of the figure

The length of the large square (the outer figure) is \(a + b\), so the area of the large square is \((a + b)^{2}\). By the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), this is the equivalent expression that uses the length of the figure to represent the area.

Step3: Derive the Pythagorean Theorem

We set the two area expressions equal: \(c^{2}+2ab=a^{2}+2ab + b^{2}\). If we subtract \(2ab\) from both sides of the equation, we get \(c^{2}=a^{2}+b^{2}\) (or \(a^{2}+b^{2}=c^{2}\)).

Answer:

1. The expression \(c^{2}+2ab\) represents the area of the figure as the sum of the area of the shaded triangles and the area of the white square.
2. The equivalent expressions \((a + b)^{2}\) (or \(a^{2}+2ab + b^{2}\)) use the length of the figure to represent the area.
3. Setting two of these area expressions equal to each other and subtracting \(2ab\) from both sides of the equation results in the Pythagorean Theorem, \(a^{2}+b^{2}=c^{2}\).