Explain a proof of the Pythagorean Theorem using the image

Answer 1

To prove the Pythagorean theorem using an image, we can use the concept of area.

Let's consider a right-angled triangle with all three sides labeled. The two smaller sides (adjacent and opposite to the right angle) are typically denoted as a and b, and the longest side (the hypotenuse) is denoted as c.

Now, we draw squares on each side of the triangle. The squares are constructed such that each side of the triangle is a side of a square.

Since the side lengths of squares on the sides a and b are equal to a and b respectively, the area of the squares will be a² and b².

Next, we arrange the three squares adjacent to each other, forming a larger square. The length of the larger square will be equal to the sum of the side lengths of the smaller squares, i.e., a + b.

Looking at the image, we notice that the larger square can be divided into four distinct regions: the two smaller squares on sides a and b, the square on the hypotenuse (denoted as c²), and two congruent right-angled triangles formed by the hypotenuse and sides a and b.

Now, we can see that the area of the larger square is equal to the sum of the areas of the smaller squares, plus the area of the two triangles.

Mathematically, this can be expressed as:
(a + b)² = a² + b² + c²

Simplifying the equation, we get:
a² + 2ab + b² = a² + b² + c²

Canceling out a² and b² from both sides, we are left with:
2ab = c²

Dividing both sides by 2, we get:
ab = c²/2

Finally, multiplying both sides by 2, we have:
2ab = c², which is the same as the Pythagorean theorem:
a² + b² = c²

Hence, we have proven the Pythagorean theorem using the image by visually demonstrating how the squares on the sides of a right-angled triangle relate to the area of the square on the hypotenuse.