Use these abrevations for the properties of real numbers and complete the flow diagram can you me?

For the first bubble or circle to fill in, the answer is Cx. So you have the proper answer here. Nice work. The pq turns into qp by the commutative property of multiplication

The answer to the second bubble is Cx for similar reasons as before. Now we have xy turn into yx

The third bubble's answer is C+ (commutative property of addition) because qp+yx swaps to yx+qp

The answer for the fourth bubble is D. You are correct.

The answer for the last (fifth) bubble is Ax to indicate "associative property of multiplication". We are simply regrouping terms when multiplying

In summary the answers from left to right are: Cx, Cx, C+, D, Ax

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240