To explain a proof of the Pythagorean Theorem using a geometric image, consider a square constructed on each side of a right triangle. Let the sides of the triangle be \(a\), \(b\), and the hypotenuse \(c\). In this configuration, we can create a large square with a side length of \(a + b\), which has an area of \((a + b)^2\). Inside this large square, we can fit the right triangle four times, with the right angles positioned at the corners of the square. The area occupied by the four triangles is \(4 \times \frac{1}{2}ab = 2ab\). The remaining area, which is the area of the smaller square formed by the hypotenuse, is \(c^2\). The area of the large square can also be expressed as the sum of the area of the smaller square and the area of the four triangles. Thus, we have the equation \((a + b)^2 = c^2 + 2ab\). Expanding the left side gives \(a^2 + 2ab + b^2\), allowing us to rearrange the equation to derive \(c^2 = a^2 + b^2\). This proves the Pythagorean Theorem, showing that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Use the image to answer the question. Explain a proof of the Pythagorean Theorem using the image. (4 points) Write the correct answer in paragraph form please. Thank you!!
3 answers
To explain a proof of the Pythagorean Theorem using a simple image, imagine a right triangle with sides \(a\) and \(b\), and the hypotenuse \(c\). Now, picture a large square that has a side length of \(a + b\). The area of this square is \((a + b)^2\). Inside this square, you can fit four copies of the right triangle. The area of those four triangles is \(2ab\). The remaining area, which is a smaller square, has a side length of \(c\) and an area of \(c^2\).
So, if you add the area of the triangles to the area of the smaller square, you get the area of the large square: \((a + b)^2 = c^2 + 2ab\). When you expand \((a + b)^2\), it becomes \(a^2 + 2ab + b^2\). This leads to the equation \(c^2 = a^2 + b^2\), proving that in a right triangle, the square of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\).