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: Calculate area of shaded triangles in Figure 1

There are 4 right - angled triangles in Figure 1. The area of a single right - angled triangle with legs \(a\) and \(b\) is \(A_{triangle}=\frac{1}{2}ab\). So the combined area of 4 such triangles is \(4\times\frac{1}{2}ab = 2ab\).

Step2: Calculate area of shaded triangles in Figure 2

There are 4 right - angled triangles in Figure 2. The area of a single right - angled triangle with legs \(a\) and \(b\) is \(A_{triangle}=\frac{1}{2}ab\). So the combined area of 4 such triangles is \(4\times\frac{1}{2}ab=2ab\). So the combined area of the shaded triangles in Figure 1 is equal to the combined area of the shaded triangles in Figure 2.

Step3: Find area of un - shaded square in Figure 1

The side length of the un - shaded square in Figure 1 is \(c\), so its area is \(c^{2}\).

Step4: Find combined area of un - shaded squares in Figure 2

The side lengths of the two un - shaded squares in Figure 2 are \(a\) and \(b\) respectively. The combined area of the two un - shaded squares is \(a^{2}+b^{2}\).

Step5: Prove Pythagorean theorem

Since the total area of the large square in both figures is the same, and the combined area of the shaded triangles is the same in both figures, we have \(a^{2}+b^{2}=c^{2}\).

Answer:

1. equal to
2. \(c^{2}\)
3. \(a^{2}+b^{2}\)
4. \(a^{2}+b^{2}=c^{2}\)