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.

Question

explain a proof of the pythagorean theorem using the image

Answer 1

To explain a proof of the Pythagorean Theorem using the described configuration of the square and the right-angled triangles, we can analyze the geometric construction step by step.

Configuration Recap:

1. We have a large outer square formed by positioning four identical right-angled triangles at each corner. Each triangle has a height $a$ and a base $b$.
2. The right angle of each triangle is at the corners of the outer square.
3. The inner square formed by the arrangement of the triangles has its vertices at points where the sides of the triangles meet the outer square, and each side of the inner square is the hypotenuse $c$ of the triangles.

Proof of the Pythagorean Theorem: The Pythagorean Theorem states that in a right-angled 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. Mathematically, this can be expressed as: $$c^2=a^2+b^2$$

Steps:

1. Understanding Triangle Properties: Each of the four right-angled triangles has legs of length $a$ and $b$, and by the definition of the right triangle: $$c=\sqrt{a^2+b^2}$$

2. Area of Shapes:

   • Calculate the area of the outer square. Since each side of the outer square is $a+b$: $$\text{Area of outer square}=(a+b{)}^2=a^2+2ab+b^2$$

   • Calculate the area of the inner square. Since each side of the inner square is $c$: $$\text{Area of inner square}=c^2=a^2+b^2$$

3. Shaded Areas: Since the area of the triangles is subtracted from the area of the outer square to find the area of the inner square, we note:

   • The area of one triangle is $\frac{1}{2}ab$.
   • The total area of the four triangles is $4\cdot \frac{1}{2}ab=2ab$.

4. Set Up the Equation: The area of the outer square must equal the area of the inner square plus the total area of the four triangles: $$(a+b{)}^2=c^2+2ab$$ Expanding the left side: $$a^2+2ab+b^2=c^2+2ab$$

5. Cancelling Terms: We can subtract $2ab$ from both sides: $$a^2+b^2=c^2$$

Conclusion:

We have derived that $c^2=a^2+b^2$ using the relationships between the areas of the outer square, inner square, and the triangular regions. This geometrical configuration not only demonstrates the areas but also visually reinforces the concept that the square of the hypotenuse (length $c$) is equal to the sum of the squares of the other two sides (lengths $a$ and $b$), thereby proving the Pythagorean Theorem.

Answer 2

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Answer 3

The Pythagorean Theorem states $c^2=a^2+b^2$ using triangle areas in a square configuration.