My First Blog Post

Welcome to my blog!

This is the content of my first blog post.

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. In mathematical terms, if a right-angled triangle has side lengths

c2=a2+b2c^2 = a^2 + b^2

Consider a right-angled triangle with sides aa, bb, and hypotenuse cc. To prove the Pythagorean Theorem, we'll use the following construction:

  1. Construct a square with side length a+ba + b, and divide it into smaller shapes:

    • The large square has an area of (a+b)2(a + b)^2.
    • Inside this square, place four identical right-angled triangles, each with sides aa, bb, and hypotenuse cc. These four triangles will form a smaller square in the center.

  2. The area of the large square is (a+b)2(a + b)^2, and the area of the four triangles is 4×12ab=2ab4 \times \frac{1}{2}ab = 2ab.

  3. The remaining area in the center is a smaller square with side length cc, so its area is c2c^2.

Thus, the total area of the large square can be expressed as the area of the four triangles plus the area of the smaller square:

(a+b)2=4×12ab+c2 (a + b)^2 = 4 \times \frac{1}{2}ab + c^2

Simplifying:

(a+b)2=2ab+c2 (a + b)^2 = 2ab + c^2

Now, expand (a+b)2(a + b)^2:

a2+2ab+b2=2ab+c2 a^2 + 2ab + b^2 = 2ab + c^2

Canceling the 2ab2ab terms from both sides:

a2+b2=c2 a^2 + b^2 = c^2

This completes the proof of the Pythagorean Theorem.