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
( i get confused at the (a-b)^2 )
(a-b)^2 + (4 a b/2)
= a^2 - 2 a b + b^2 + 2 a b
= a^2 + b^2
but as you know
c^2 = a^2 + b^2
is the Pythagorean Theorem
Look at this
scroll down to "algebraic proofs"
http://en.wikipedia.org/wiki/Pythagorean_theorem
THANKS AGAIN
To simplify the equation c^2 = (a-b)^2 + (4 * ab/2), let's break it down step by step.
1. Start with the equation c^2 = (a-b)^2 + (4 * ab/2).
2. Simplify the expression (4 * ab/2) by multiplying 4 and ab and then dividing by 2:
(4 * ab/2) = 2ab
The expression (4 * ab/2) simplifies to 2ab.
3. Rewrite the equation with the simplified expression:
c^2 = (a-b)^2 + 2ab
4. Expand (a-b)^2 by multiplying (a-b) with itself:
(a-b)^2 = (a-b)(a-b) = a^2 - 2ab + b^2
When we expand (a-b)^2, we use the distributive property to multiply (a-b) with (a-b).
5. Substitute the expanded expression of (a-b)^2 back into the equation:
c^2 = a^2 - 2ab + b^2 + 2ab
We substitute (a^2 - 2ab + b^2) back into the equation since (a-b)^2 equals a^2 - 2ab + b^2.
6. Combine like terms:
c^2 = a^2 - 2ab + 2ab + b^2
We combine -2ab and +2ab since they cancel each other out.
7. Simplify further:
c^2 = a^2 + b^2
The terms -2ab and 2ab completely cancel out, leaving us with a simplified equation.
Therefore, c^2 = a^2 + b^2, which is known as the Pythagorean Theorem.
The Pythagorean Theorem states that in a right-angled 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).