To explain a proof of the Pythagorean Theorem using the described configuration of the square and the right-angled triangles, we can analyze the geometric construction step by step.
Configuration Recap:
- We have a large outer square formed by positioning four identical right-angled triangles at each corner. Each triangle has a height
and a base . - The right angle of each triangle is at the corners of the outer square.
- The inner square formed by the arrangement of the triangles has its vertices at points where the sides of the triangles meet the outer square, and each side of the inner square is the hypotenuse
of the triangles.
Proof of the Pythagorean Theorem:
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as:
Steps:
-
Understanding Triangle Properties: Each of the four right-angled triangles has legs of length
and , and by the definition of the right triangle: -
Area of Shapes:
-
Calculate the area of the outer square. Since each side of the outer square is
: -
Calculate the area of the inner square. Since each side of the inner square is
:
-
-
Shaded Areas: Since the area of the triangles is subtracted from the area of the outer square to find the area of the inner square, we note:
- The area of one triangle is
. - The total area of the four triangles is
.
- The area of one triangle is
-
Set Up the Equation: The area of the outer square must equal the area of the inner square plus the total area of the four triangles:
Expanding the left side: -
Cancelling Terms: We can subtract
from both sides:
Conclusion:
We have derived that