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A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.

Explain a proof of the Pythagorean Theorem using the image.

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To prove the Pythagorean Theorem using the given figure, we can visually interpret the arrangement of the right-angled triangles and the squares. Here's a step-by-step explanation of the proof:

  1. Understanding the Configuration: We have a large outer square and an inner square formed by the placement of four right-angled triangles. Each triangle has legs of length \(a\) and \(b\), and the hypotenuse of each triangle, denoted as \(c\), acts as a side of the inner square.

  2. Calculating the Area of the Outer Square: The side length of the outer square is the sum of the segments \(a\) and \(b\) from two triangles. Thus, the side length \(L\) of the outer square can be given as: \[ L = a + b \] The area of the outer square, therefore, is: \[ \text{Area}_{\text{outer}} = L^2 = (a + b)^2 = a^2 + 2ab + b^2 \]

  3. Calculating the Area of the Inner Square: The side length of the inner square is denoted \(c\). The area of the inner square is calculated as: \[ \text{Area}_{\text{inner}} = c^2 \]

  4. Establishing Relationships: Since the four right-angled triangles occupy the corners of the outer square, we can analyze the area of the outer square in terms of the inner square's area and the triangles’ area. There are four right-angled triangles in total. The area of one triangle is given by: \[ \text{Area}{\text{triangle}} = \frac{1}{2}ab \] Therefore, the total area of the four triangles is: \[ \text{Area}{\text{triangles}} = 4 \times \frac{1}{2}ab = 2ab \]

  5. Setting Up the Equation: The area of the outer square can be expressed as the sum of the area of the inner square and the area of the four triangles: \[ \text{Area}{\text{outer}} = \text{Area}{\text{inner}} + \text{Area}_{\text{triangles}} \] Substituting the areas we calculated gives: \[ a^2 + 2ab + b^2 = c^2 + 2ab \]

  6. Simplifying the Equation: By subtracting \(2ab\) from both sides, we simplify the equation to: \[ a^2 + b^2 = c^2 \]

  7. Conclusion: This final equation \(a^2 + b^2 = c^2\) is the statement of the Pythagorean Theorem. Therefore, through this visual representation and breakdown of areas, we have derived the Pythagorean Theorem using the properties of squares and right-angled triangles.

This proof illustrates how the arrangement of the triangles within the squares can lead us directly to the conclusion of the Pythagorean Theorem.