Question

use the drop-down menus to complete the proof of the pythagorean theorem using the figures. click the arrows to choose an answer from each menu. the combined area of the shaded triangles in figure 1 is choose... the combined area of the shaded triangles in figure 2. the area of the unshaded square in figure 1 can be represented by choose... . the combined area of the two unshaded squares in figure 2 can be represented by choose... . the areas of the squares in figure 1 and figure 2 show that choose... .

Explanation:

Step1: Analyze shaded triangles

In Figure 1, there are 4 right - angled triangles with legs \(a\) and \(b\) and hypotenuse \(c\). The area of each triangle is \(\frac{1}{2}ab\), so the combined area of shaded triangles in Figure 1 is \(4\times\frac{1}{2}ab = 2ab\). In Figure 2, there are also 2 right - angled triangles (the two orange triangles) with legs \(a\) and \(b\) and 2 other right - angled triangles? Wait, no. Wait, actually, in Figure 2, the shaded regions: let's count the number of triangles. The first orange region is a right - angled triangle with legs \(a\) and \(b\), the second orange region is also a right - angled triangle with legs \(a\) and \(b\), and then there are two other triangles? Wait, no, actually, the key is that the number of triangles with area \(\frac{1}{2}ab\) in Figure 1 is 4, and in Figure 2, if we look at the shaded parts, the two orange triangles and the other two? Wait, no, actually, the combined area of the shaded triangles in Figure 1 and Figure 2: in Figure 1, 4 triangles each with area \(\frac{1}{2}ab\), so total \(2ab\). In Figure 2, the shaded triangles: the top orange triangle has area \(\frac{1}{2}ab\), the bottom orange triangle has area \(\frac{1}{2}ab\), and are there two more? Wait, no, maybe I made a mistake. Wait, actually, the combined area of the shaded triangles in Figure 1 is equal to the combined area of the shaded triangles in Figure 2. Because in Figure 1, we have 4 triangles, and in Figure 2, we can also find that the total area of the shaded triangles is the same. So the first blank: "equal to".

Step2: Unshaded square in Figure 1

The unshaded square in Figure 1 has side length \(c\), so its area is \(c^{2}\).

Step3: Combined area of unshaded squares in Figure 2

In Figure 2, the unshaded squares have side lengths \(a\) and \(b\) respectively. So their combined area is \(a^{2}+b^{2}\).

Step4: Pythagorean Theorem conclusion

Since the area of the large square (which is \((a + b)^{2}\)) in both figures is the same, and the area of the shaded triangles is the same in both figures, the area of the unshaded square in Figure 1 (\(c^{2}\)) must be equal to the combined area of the unshaded squares in Figure 2 (\(a^{2}+b^{2}\)). So \(a^{2}+b^{2}=c^{2}\).

Answer:

1. The combined area of the shaded triangles in Figure 1 is \(\boldsymbol{\text{equal to}}\) the combined area of the shaded triangles in Figure 2.
2. The area of the unshaded square in Figure 1 can be represented by \(\boldsymbol{c^{2}}\).
3. The combined area of the two unshaded squares in Figure 2 can be represented by \(\boldsymbol{a^{2}+b^{2}}\).
4. The areas of the squares in Figure 1 and Figure 2 show that \(\boldsymbol{a^{2}+b^{2}=c^{2}}\) (the Pythagorean Theorem).