The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed in the formula:
\[ c^2 = a^2 + b^2 \]
where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two sides.
One common proof of the Pythagorean Theorem is done using a geometric approach, often referred to as the "square proof." Here’s how it works:
Proof:
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Start with a Right Triangle: Consider a right triangle \( \triangle ABC \) where \( \angle C \) is the right angle. Let side \( AC = b \), side \( BC = a \), and the hypotenuse \( AB = c \).
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Create a Square: Construct a square on each side of the triangle. The large square will have a side length of \( a + b \), and thus an area of \((a + b)^2\).
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Arrange Four Triangles: Inside this large square, place four identical right triangles (all \( \triangle ABC \)) in each corner of the square. The area of one triangle is \(\frac{1}{2}ab\), so the area of the four triangles combined is \(4 \times \frac{1}{2}ab = 2ab\).
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Find the Area of the Large Square: The area of the large square can also be calculated in two ways:
- Directly, as \((a + b)^2\).
- By subtracting the area of the four triangles from the area of the whole square. The remaining area is the area of the inner square, which has a side length of \(c\). Thus its area is \(c^2\).
So, combining these: \[ \text{Area of large square} = \text{Area of inner square} + \text{Area of four triangles} \] This can be expressed as: \[ (a + b)^2 = c^2 + 2ab \]
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Expand and Rearrange: Expanding the left-hand side gives: \[ a^2 + 2ab + b^2 = c^2 + 2ab \] Now, subtract \(2ab\) from both sides: \[ a^2 + b^2 = c^2 \]
Thus, we have shown that \(c^2 = a^2 + b^2\), which proves the Pythagorean Theorem.
Conclusion:
This geometric proof visually demonstrates the relationship between the areas of the squares constructed on the sides of a right triangle, supporting the theorem that relates the lengths of the sides of a right triangle. Various other proofs exist (including algebraic and trigonometric proofs), but this one is particularly helpful for illustrating the relationship geometrically.