Question

Greg drew a square with each side equal to z units.

Part A: Greg increased the length and width of the square by t units each. What will be the change in the area of the original figure? Show your work.

Part B: Greg decreased the length and width of the square by t units each. What will be the change in the area of the square? Show your work.

Part C: Greg increased the length of the square by t units and decreased its width by t units. What will be the change in the area of the square? Show your work.

Answer 1

Part A)
Original square = z^2
New square = tz^2

Change in the area = (tz^2) - (z^2)
(a – b)^2 = a^2 – 2ab + b^2

Answer 2

So far I have that, and I am not sure if that is right.

Answer 3

I don't think that is correct.
In Part A,
he did not increase the side length by a factor of t.
He increased it by t.
That means the new square has dimensions z + t by z + t, not zt by zt.

Answer 4

Ohhh okay :) Thank you! But how would I find the change in the area?

Answer 5

The new equation would be (t + z)^2

Answer 6

Part A:
Area of original square is: A1 = z^2
Area of larger square is: A2 = (z + t)^2 = (z + t)(z + t)

Now you need to multiply out (z + t)(z + t) using FOIL, or perhaps you remember that the square of binomial follows this pattern:
\((a + b)^2 = a^2 + 2ab + b^2\)

Answer 7

Wouldn't it be subtracting, since you are trying to find the change?

Answer 8

Yes. Correct. First multiply out (z + t)^2.
Then subtract z^2 from it.

Answer 9

Okay, so wouldn't we use (a – b)^2 = a^2 – 2ab + b^2 that equation instead?

Answer 10

Aren't we still working on Part A?
In part A, the pattern is (a + b)^2 = a^2 + 2ab + b^2 since (z + t)^2 has a plus sign inside the parentheses, not a minus sign.

Answer 11

Oh, okay! Thank you!

Answer 12

Once I get that, I would use the other equation to find the change?

Answer 13

For Part B, the side of the square becomes z - t, so the area of the smaller new square will be (z - t)^2, and it will follow the pattern you showed above, (a - b)^2 = a^2 - 2ab + b^2. Once you have the new area, you once again will subtract the new area from the old area.

Answer 14

Finally, for Part C, the length and width of the square are not both increased or decreased by the same amount, so the new figure is no longer a square. In this case to find the new area you must use FOIL. Then once again, you subtract the old area from the new area to find the change in the area.

Answer 15

Thank you so much! :3