Use the image to answer the question.
A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.
Explain a proof of the Pythagorean Theorem using the image
To prove the Pythagorean Theorem using the given image, we can focus on the two smaller right-angled triangles formed inside the outer square. These triangles share the same leg, labeled a, and the same hypotenuse, labeled c.
From the image, we can observe that the longer leg of each small triangle, labeled b, connects two of the vertices of the inner square. As mentioned earlier, each side of the inner square is also the hypotenuse of a right-angled triangle formed at the corners of the outer square.
Since both small triangles share the same hypotenuse c, we can conclude that the sum of the squares of their legs (a^2 + b^2) is equal to the square of the hypotenuse (c^2) according to the Pythagorean Theorem.
Moreover, since the two small triangles are congruent (having the same dimensions), their corresponding legs (a and b) are equal in length. Therefore, we can replace both a and b with x, making the formula x^2 + x^2 = c^2 simpler to write, which further demonstrates the Pythagorean Theorem.
In conclusion, the given image provides a visual representation of the Pythagorean Theorem by showing how the squares of the legs of right-angled triangles (a^2 + b^2) are equal to the square of the hypotenuse (c^2). This proof is based on the congruence of the two smaller right-angled triangles within the outer square.