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.

Question

Explain a proof of the Pythagorean Theorem using the image. 5 grade words

Answer 1

To explain a proof of the Pythagorean Theorem using the figure you described, let's break it down step by step.

1. Understanding the Shapes: We have an outer square and an inner square made by four right-angled triangles. The outer square is large, and the inner square is smaller. Each triangle has a height (a) and a base (b). The hypotenuse of each triangle is labeled (c), which is also the side length of the inner square.

2. Setting Up the Problem: In the outer square, each side is made up of the shorter part (a) and the longer part (b). So the total length of each side of the outer square is a + b + a + b = 2a + 2b.

3. Relating Right Triangles to the Squares: When we look at each triangle, we can see that we can use these sides (a and b) to form the triangles. The triangles are arranged in a way that their right angles fit nicely into the corners of the outer square.

4. Using the Pythagorean Theorem: In each triangle, if we look at the right-angled triangle, we can use the Pythagorean Theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). This can be written as: \[ c^2 = a^2 + b^2 \]

5. Finding the Area of Squares: The area of the outer square can be calculated as the side length squared: \[ (\text{Side of Outer Square})^2 = (2a + 2b)^2 \] Simplifying this gives: \[ (2a + 2b)(2a + 2b) = 4a^2 + 8ab + 4b^2 \]

6. Finding the Area of the Inner Square: The area of the inner square is simply: \[ c^2 \]

7. Subtracting the Area of Triangles: The area of the four shaded triangles can be calculated. The area of one triangle is: \[ \text{Area} = \frac{1}{2}ab, \] So for four triangles, we multiply by 4: \[ \text{Total Area of Triangles} = 4 \times \left(\frac{1}{2}ab\right) = 2ab \]

8. Relating Areas: Now, if we want to relate these areas:

   • The area of the outer square equals the area of the inner square plus the area of the four triangles: \[ (2a + 2b)^2 = c^2 + 2ab \]

9. Putting It All Together: We already showed that \( c^2 = a^2 + b^2 \). If we combine these findings, we can confirm the Pythagorean Theorem: \[ 4a^2 + 8ab + 4b^2 = a^2 + b^2 + 2ab \] This shows that the relationships between the sides a, b, and c in right triangles confirm the Pythagorean Theorem.

So, using the areas and triangles, we can prove that \( c^2 = a^2 + b^2 \), which is the Pythagorean Theorem!

Answer 2

shorter

Answer 3

Sure! Here’s a shorter explanation of the proof of the Pythagorean Theorem using the figure you described:

1. Shapes: We have an outer square and an inner square formed by four right-angled triangles. Each triangle has a height (a), a base (b), and a hypotenuse (c).

2. Triangle Relations: In each triangle, the Pythagorean Theorem states: \[ c^2 = a^2 + b^2 \]

3. Area of the Outer Square: The side of the outer square is \(2a + 2b\). So, its area is: \[ (2a + 2b)^2 = 4a^2 + 8ab + 4b^2 \]

4. Area of the Inner Square: The area of the inner square is simply \(c^2\).

5. Area of the Triangles: The four triangles have a combined area of \(2ab\) (since each triangle’s area is \(\frac{1}{2}ab\)): \[ \text{Total Area of Triangles} = 4 \times \left(\frac{1}{2}ab\right) = 2ab \]

6. Combining Areas: The area of the outer square equals the area of the inner square plus the area of the four triangles: \[ (2a + 2b)^2 = c^2 + 2ab \] This confirms that: \[ 4a^2 + 8ab + 4b^2 = a^2 + b^2 + 2ab \]

7. Conclusion: This shows that \(c^2 = a^2 + b^2\), proving the Pythagorean Theorem using the shapes!

Answer 4

super shorter

Answer 5

Certainly! Here’s a super short version:

1. Shapes: In the figure, we have an outer square and an inner square formed by four right triangles with sides \(a\) (height), \(b\) (base), and \(c\) (hypotenuse).

2. Pythagorean Theorem: In each triangle, the relation is: \[ c^2 = a^2 + b^2 \]

3. Area of Squares: The area of the outer square is \((2a + 2b)^2 = 4a^2 + 8ab + 4b^2\).

4. Triangle Area: The combined area of the four triangles is \(2ab\).

5. Area Equation: The area of the outer square equals the area of the inner square plus the area of the triangles: \[ 4a^2 + 8ab + 4b^2 = c^2 + 2ab \]

6. Conclusion: This confirms \(c^2 = a^2 + b^2\), proving the Pythagorean Theorem!