In order to find the answer to this one, we're going to need to do a lot of algebra. First of all, remember that a rectangle has two sets of congruent sides, which make the perimeter when put together. I'll define the two different measurements (one for one set, one for the other set) as l and w. Therefore, 2l + 2w = P, or the two sets of congruent sides equal the perimeter.
In this problem, l = x + 12 and w = 2x + 8. The perimeter is 18x - 20. Let's plug these values into our earlier equation.
2(x + 12) + 2(2x + 8) = 18x - 20 Substitution
2x + 24 + 4x + 16 = 18x - 20 Distribute
6x + 40 = 18x - 20 Combine like terms
6x = 18x - 60 Subtract 40 from both sides
-12x = -60 Subtract 18x from both sides
x = 5 Divide both sides by -12
Therefore, x = 5. Let's plug that in to l (x + 12) and w (2x + 8) to find the side lengths.
5 + 12 Substitution
17 Add
The length of one side is 17 units.
2(5) + 8 Substitution
10 + 8 Multiply
18 Add
The length of the other side is 18 units.
Now, let's plug 5 into the perimeter (18x - 20).
18(5) - 20 Substitution
90 - 20 Multiply
70 Subtract
Therefore, the perimeter is 70 units. We can check this by plugging into the earlier equation we made to find x.
2l + 2w = 70 Given
2(17) + 2(18) = 70 Substitute each side length
34 + 36 = 70 Multiply
70 + 70 Add
Since the equation is true, our math checks out. Our answer is correct!
Therefore, x = 5, the side lengths are 17 and 18 units, and the perimeter is 70 units.
Hope this helps!
