Question

the diagram shows a larger square divided into a smaller square and four right triangles. the diagram can be used to prove the pythagorean theorem. complete the proof of the pythagorean theorem. the area of the larger square found by multiplying its side lengths is, which is equivalent to ( a^{2}+2ab + b^{2} ). the combined area of the four triangles and the smaller square is. these two expressions both represent the same area, so ( a^{2}+2ab + b^{2}=2ab + c^{2} ). finally, subtract from both sides of the equation. therefore, ( sqrt{a^{2}+b^{2}}=c^{2} ) ( a^{2}+b^{2}=c^{2} )

Explanation:

Step1: Find the area of the larger square

The side length of the larger square is \(a + b\). Using the formula for the area of a square \(A=s^{2}\) (where \(s\) is the side - length), the area is \((a + b)^{2}\). Expanding \((a + b)^{2}\) using the formula \((x+y)^{2}=x^{2}+2xy + y^{2}\) (here \(x = a\) and \(y = b\)), we get \(a^{2}+2ab + b^{2}\).

Step2: Find the combined area of the four triangles and the smaller square

The area of a right - triangle is \(A=\frac{1}{2}xy\) (for a right - triangle with legs \(x\) and \(y\)). Each of the four right - triangles has legs \(a\) and \(b\), so the area of one triangle is \(\frac{1}{2}ab\), and the area of four triangles is \(4\times\frac{1}{2}ab=2ab\). The side length of the smaller square is \(c\), so its area is \(c^{2}\). The combined area of the four triangles and the smaller square is \(2ab + c^{2}\).

Step3: Solve the equation

We have the equation \(a^{2}+2ab + b^{2}=2ab + c^{2}\). Subtract \(2ab\) from both sides of the equation.

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Answer:

The area of the larger square found by multiplying its side lengths is \((a + b)^{2}\), the combined area of the four triangles and the smaller square is \(2ab + c^{2}\), subtract \(2ab\) from both sides of the equation. Therefore, \(a^{2}+b^{2}=c^{2}\).