Use the image to answer the question.

Question

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.

(4 points)

Math Short Answer Rubric (4 points)
This table shows the criteria for awarding points and the corresponding point values.
Points Awarded Criteria
4
The answer is clear and correct.

The answer shows that the student used problem-solving skills and reasoning.

All work is shown or explained.

3
The answer is mostly correct. The student made small errors while solving.

The answer shows that the student used some problem-solving skills and reasoning.

Most work is shown or explained.

2
The answer is partially correct. The student made errors while solving.

The answer shows that the student used some problem-solving skills or reasoning.

Some work is shown or explained.

1
The answer is incorrect. The student made errors while solving.

The answer shows little problem-solving skills or reasoning.

Little or no work is shown.

0 The question is not answered.


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Answer 1

To prove the Pythagorean Theorem using the described image of a square made from four right-angled triangles, we can follow these steps:

1. Understanding the Geometry: We have an outer square formed by the four right-angled triangles, each with a height (a), base (b), and hypotenuse (c). The triangles are arranged such that their right angles form the corners of the outer square. The squares' inner sides of the formed square (which is diagonal to the outer square) will have length c, which serves as the hypotenuse of each triangle.

2. Area of the Outer Square: The area of the outer square can be calculated by taking the side length, which is (a + b). Therefore, the area \(A_{outer} = (a + b)^2\).

3. Area of Four Triangles: Next, we calculate the total area of the four right-angled triangles. The area of one triangle is given by the formula \(A_{triangle} = \frac{1}{2} \times base \times height = \frac{1}{2} \times b \times a\). Thus, the total area of the four triangles is: \[ A_{triangles} = 4 \times \left(\frac{1}{2} \times b \times a\right) = 2ab \]

4. Area of the Inner Square: The area of the inner square, where each side is the hypotenuse (c), is given by \(A_{inner} = c^2\).

5. Relation Between Areas: The area of the outer square is equal to the area of the inner square plus the total area of the four triangles: \[ A_{outer} = A_{inner} + A_{triangles} \] Substituting the area calculations gives us: \[ (a + b)^2 = c^2 + 2ab \]

6. Expanding the Left Side: Expanding the left-hand side: \[ (a + b)^2 = a^2 + 2ab + b^2 \]

7. Setting Up the Equation: We can set the two expressions for the area equal to each other: \[ a^2 + 2ab + b^2 = c^2 + 2ab \]

8. Simplifying: If we subtract \(2ab\) from both sides, we get: \[ a^2 + b^2 = c^2 \]

9. Conclusion: This equation \(a^2 + b^2 = c^2\) represents the Pythagorean Theorem, demonstrating that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Thus, the construction of the outer square and the arrangement of the triangles provide a geometric proof of the Pythagorean Theorem.