Triangles Unit Test

To prove the Pythagorean Theorem using the described square and right-angled triangles, we can follow these steps:

1. Set Up the Image: We start with a large square made up of four identical right triangles placed at the corners and an inner square formed by the vertices of the triangles. Let’s denote the sides of the outer square as \(a + b\) (where \(a\) is the shorter side and \(b\) is the longer side) and the side of the inner square, which is also the hypotenuse of the right triangles, as \(c\).

2. Identify the Dimensions of the Triangles: Each of the four right triangles has:

   • Height (perpendicular side) = \(a\)
   • Base (adjacent side) = \(b\)
   • Hypotenuse = \(c\)

3. Base of Large Square: The side length of the larger square, incorporating all four triangles, is \(a + b\). Since the triangles are arranged in all four corners, it can be confirmed that this is the sum of the two segments (one from each triangle) at the vertices.

4. Area of Squares:

   • The area of the outer square is \((a + b)^2\).
   • The area of the inner square (formed by the hypotenuses of the triangles) is \(c^2\).

5. Area of Triangles: Each right triangle has an area given by the formula: \[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \] Thus, the total area of the four triangles is: \[ 4 \times \frac{1}{2} \times a \times b = 2ab \]

6. Expressing Areas in Relation to Each Other: The area of the outer square can also be expressed as: \[ \text{Area of outer square} = \text{Area of inner square} + \text{Area of four triangles} \] Therefore, \[ (a + b)^2 = c^2 + 2ab \]

7. Expand the Square: Expanding the left side gives: \[ a^2 + 2ab + b^2 = c^2 + 2ab \]

8. Cancel Out Common Terms: We can subtract \(2ab\) from both sides: \[ a^2 + b^2 = c^2 \]

9. Conclusion - The Pythagorean Theorem: The final equation \(a^2 + b^2 = c^2\) is precisely the statement of 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 \(b\)).

This proof not only gives a clear relation between the triangles' dimensions but also visually demonstrates how these relationships hold within geometric figures.