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Use the figure to prove the Pythagorean Theorem. Assume that segments appearing to be perpendicular form right angles. Make proof table

In the given image, we observe a square at the center and a surrounding quadrilateral composed of right-angled triangles. We can use this configuration to construct a proof of the **Pythagorean Theorem** by comparing the areas.

### Step-by-step Proof Using a Proof Table

| **Step** | **Explanation** |
|:--------|:----------------|
| **1. Identify the shapes.** | There is a central square, and four right-angled triangles are arranged around this square. The right triangles have sides of length \( a \), \( b \), and hypotenuse \( c \). |
| **2. Area of the large square.** | The outer square is formed by combining the four right triangles and the inner square. The side of the outer square is \( a + b \), so its area is \( (a + b)^2 \). |
| **3. Area of the inner square.** | The inner square, which is formed by the hypotenuses of the right triangles, has side length \( c \). Therefore, the area of the inner square is \( c^2 \). |
| **4. Total area in two ways.** | The total area of the large square can be found in two different ways: 1) as the sum of the areas of the four triangles and the inner square, and 2) by directly calculating the area of the large square as \( (a + b)^2 \). |
| **5. Calculate the areas of the triangles.** | The area of each triangle is \( \frac{1}{2}ab \), and since there are 4 triangles, their combined area is \( 4 \times \frac{1}{2}ab = 2ab \). |
| **6. Compare the two area expressions.** | The total area is \( (a + b)^2 \) and can also be expressed as the sum of the inner square's area and the triangles' area: \( c^2 + 2ab \). |
| **7. Expand and simplify.** | Expanding \( (a + b)^2 \) gives \( a^2 + 2ab + b^2 \). Therefore, we have \( a^2 + 2ab + b^2 = c^2 + 2ab \). |
| **8. Subtract and conclude.** | Subtracting \( 2ab \) from both sides gives \( a^2 + b^2 = c^2 \), which is the Pythagorean Theorem. |

### Final Conclusion:
The diagram, through an area comparison approach, provides a geometric proof of the Pythagorean Theorem: \( a^2 + b^2 = c^2 \).

Would you like more details or have any questions?

Here are 5 relative questions to further expand the concept:
1. How would the proof change if the triangles were not right-angled?
2. Can this geometric method be used for triangles with different types of angles?
3. How would you prove the Pythagorean Theorem algebraically without using geometry?
4. How can this geometric method be applied to find the length of the hypotenuse if \( a \) and \( b \) are known?
5. What are some real-life applications of the Pythagorean Theorem?

**Tip:** The Pythagorean Theorem is foundational in geometry and is often a starting point for exploring trigonometry. Understanding its proofs helps with deeper concepts in mathematics.

Use the figure to prove the Pythagorean Theorem. Assume that segments appearing to be perpendicular form right angles. Make proof table.

This proof of the Pythagorean Theorem uses a geometric approach by comparing the areas of triangles and squares in the provided figure. Learn how the relationship a^2 + b^2 = c^2 is derived through area calculations. Suitable for students in Grades 8-10.

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