Andre drew four congruent triangles, with legs a and b units long and hypotenuse c units long. Then, Andre arranged the triangles to form a large square. He then wrote two different expressions for the are of the large square. Since each side measures c, he represented the area as c^2. The area of the large square is also equal to the area of the square hole, (a-b)^2, plus the area of the 4 triangles, (4* ab/2). Then he set the two expressions equal. Simplify the equation c^2 = (a-b)^2 + (4 * ab/2). SHOW ALL WORK AND EXPLAIN PLZ THEN IDENTIFY THE THEOREM IT PROVES PLZ EXPLAIN
To simplify the equation c^2 = (a-b)^2 + (4 * ab/2), let's follow these steps:
1. Expand the expression (a-b)^2:
(a-b)^2 = a^2 - 2ab + b^2
2. Simplify the expression (4 * ab/2):
(4 * ab/2) = 2ab
3. Substitute the expanded expressions back into the equation:
c^2 = a^2 - 2ab + b^2 + 2ab
4. Combine like terms:
c^2 = a^2 + b^2
Now, let's explain what this equation represents and identify the theorem it proves.
The equation c^2 = a^2 + b^2 represents the Pythagorean Theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (c^2) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2).
In this case, Andre drew four congruent right triangles with legs a and b units long and hypotenuse c units long. By arranging them to form a large square, the area of the square is represented as c^2. The area of the square hole, formed by subtracting the smaller square (with side length (b-a)) from the large square, is represented by (a-b)^2. Additionally, the area of the four triangles is represented by (4 * ab/2), which simplifies to 2ab.
By setting these two expressions equal to each other (c^2 = (a-b)^2 + 2ab), Andre demonstrates the Pythagorean Theorem once again.
To simplify the equation c^2 = (a-b)^2 + (4 * ab/2), we can follow the order of operations and simplify each term step by step.
1. Starting with (a-b)^2, we can apply the power of a binomial formula:
(a-b)^2 = a^2 - 2ab + b^2
2. Next, let's simplify (4 * ab/2):
(4 * ab/2) = 2ab
3. Now, we can rewrite the equation with the simplified terms:
c^2 = a^2 - 2ab + b^2 + 2ab
4. Since 2ab and -2ab cancel each other out, the equation can be further simplified:
c^2 = a^2 + b^2
This equation c^2 = a^2 + b^2 represents the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c^2) is equal to the sum of the squares of the other two sides (a^2 + b^2). Therefore, the simplified equation proves the Pythagorean theorem.