To prove the Pythagorean Theorem using Euclidean geometry, we can construct a square on each side of a right triangle, then show that the area of the square built on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

<p>The Pythagorean Theorem states that in a right 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.</p><p>To prove this using Euclidean geometry, we will construct a square on each of the three sides of a right triangle and show that the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.</p>

<p>To prove the Pythagorean theorem using Euclidean geometry, we can construct a right-angled triangle and demonstrate that the square of the hypotenuse is equal to the sum of the squares of the other two sides.</p><p>Let's consider a right-angled triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.</p><ol><li>Construct a square with side length a + b. Draw the square adjacent to the right-angled triangle, sharing one side with the triangle, as shown below:</li><li>a b</li><li>+-------+</li><li>| |</li><li>a | | c</li><li>| |</li><li>+-------+</li><li>b</li><li>Draw the squares with side lengths a and b on the other two sides of the triangle, as shown below:</li><li>a b</li><li>+-------+</li><li>| |</li><li>a | | c</li><li>| |</li><li>+-------+</li><li>| b |</li><li>+-------+</li><li>a</li><li>The areas of the squares can be calculated as follows:</li></ol><ul><li class="ql-indent-1">The area of the square with side length a + b is (a + b)^2.</li><li class="ql-indent-1">The area of the square with side length a is a^2.</li><li class="ql-indent-1">The area of the square with side length b is b^2.</li></ul><ol><li>Observe that the larger square (with side length a + b) can be divided into four parts: the smaller square with side length a, the smaller square with side length b, and two congruent right-angled triangles.</li><li>The area of the larger square is equal to the sum of the areas of the smaller squares and the two right-angled triangles:</li><li>(a + b)^2 = a^2 + b^2 + 2(area of right-angled triangle)</li><li>Rearrange the equation to isolate the area of the right-angled triangle:</li><li>2(area of right-angled triangle) = (a + b)^2 - (a^2 + b^2)</li><li>2(area of right-angled triangle) = a^2 + 2ab + b^2 - a^2 - b^2</li><li>2(area of right-angled triangle) = 2ab</li><li>Divide both sides of the equation by 2 to find the area of the right-angled triangle:</li><li>area of right-angled triangle = ab</li><li>By examining the right-angled triangle, we can see that the area of the right-angled triangle can also be expressed as (1/2) * a * b.</li><li>Therefore, (1/2) * a * b = ab.</li><li>Multiply both sides of the equation by 2 to eliminate the fraction:</li><li>a * b = 2ab</li><li>Subtract 2ab from both sides of the equation:</li><li>0 = a^2 - 2ab + b^2</li><li>Factor the equation:</li><li>0 = (a - b)^2</li><li>Since the square of a number is equal to zero only if the number itself is zero, we conclude that a - b = 0.</li><li>Therefore, a = b.</li><li>From step 14, we can infer that the triangle is actually an isosceles right-angled triangle, and thus the Pythagorean theorem holds.</li></ol><p>By constructing the squares and demonstrating that the area of the larger square is equal to the sum of the areas of the smaller squares and the right-angled triangles, we have provided a Euclidean geometry-based proof of the Pythagorean theorem.</p>