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.

Answer 1

To prove the Pythagorean Theorem using the given image, we can start by understanding the structure formed by the outer square and the four right-angled triangles.

1. Setup the Problem: The outer square has each side divided into two parts: \( a \) (the shorter part) and \( b \) (the longer part). Each of the vertices of the inner square lies at points where these divisions occur. Therefore, each vertex of the inner square achieves the right angles created by the dimensions of the triangles.

2. Identify the Variables:

   • Each right-angled triangle has:
     • Height = \( a \)
     • Base = \( b \)
     • Hypotenuse = \( c \)

3. Understanding the Area of Squares:

   • The area of the outer square, with side length \( a + b \), is given by: \[ \text{Area}_{\text{outer}} = (a + b)^2 \]
   • The area of the inner square, with side length \( c \), is given by: \[ \text{Area}_{\text{inner}} = c^2 \]

4. Calculate the Area of the Triangles: The area of one right-angled triangle can be calculated as: \[ \text{Area}{\text{triangle}} = \frac{1}{2} \times a \times b \] Therefore, the total area of the four triangles is: \[ \text{Area}{\text{triangles}} = 4 \times \left(\frac{1}{2} \times a \times b\right) = 2ab \]

5. Relating Areas: Now, we can express the area of the outer square in terms of the area of the inner square and the total area of the triangles. The relationship can be given as: \[ \text{Area}{\text{outer}} = \text{Area}{\text{inner}} + \text{Area}_{\text{triangles}} \] Substituting what we found: \[ (a + b)^2 = c^2 + 2ab \]

6. Expand and Simplify: Expanding the left side gives: \[ a^2 + 2ab + b^2 = c^2 + 2ab \] Now, subtract \( 2ab \) from both sides: \[ a^2 + b^2 = c^2 \]

7. Conclusion: This final equation \( a^2 + b^2 = c^2 \) is the statement of the Pythagorean Theorem, confirming the relationship between the sides of right-angled triangles formed in the square construction.

Thus, the image and the arrangement described provide a geometric interpretation of the Pythagorean Theorem, solidifying its proof through the analysis of areas and their relationships.

Answer 2

short it and put it all in one sentence with no extra details

Answer 3

By arranging four identical right-angled triangles with legs \( a \) and \( b \) to form an outer square of side \( a+b \) and an inner square with hypotenuse \( c \), we can derive the equation \( (a+b)^2 = c^2 + 2ab \), which simplifies to \( a^2 + b^2 = c^2 \), thereby proving the Pythagorean Theorem.

Answer 4

short it to here to----------------------------here

Answer 5

By arranging four identical right-angled triangles with legs \( a \) and \( b \) to form an outer square of side \( a+b \) and an inner square with hypotenuse \( c \), we can derive the equation \( (a+b)^2 = c^2 + 2ab \), which simplifies to \( a^2 + b^2 = c^2 \).