10. D. 5 1/3
11. C. 285 cm
12. A. 42 units
Step-by-step explanation:
The first two are simple addition problems. (In problem 11, you can replace the addition of 5 identical numbers with a multiplication by 5.) The last problem is also an addition problem, but figuring out what to add can take a little effort.
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10. The segment addition theorem tells you that for R between S and T:
ST = SR + RT
ST = 3 2/3 + 1 2/3 = (3 +1) + (2/3 +2/3) = 4 + 4/3 = 4 + 1 1/3
ST = 5 1/3
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11. The definition of a regular polygon is that all of its sides and angles are congruent. A "pentagon" is 5-sided polygon, so your regular pentagon will have 5 sides, each of measure 57 cm. The perimeter is the total length of the sides, so is ...
P = 57 cm +57 cm +57 cm +57 cm +57 cm
P = 5×57 cm
P = 285 cm
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12. Again, the perimeter is the sum of the side lengths. Here, the length of the top side is easily figured by the difference of x-coordinates (6 units). The length of the left side is recognizable as double the length of the hypotenuse of a 3-4-5 right triangle (10 units). The lengths of the other two sides can be found using the distance formula with the end point coordinates:
MN = √((-10-8)^2 +(-6-(-3))^2) = √(324 +9) = √333 ≈ 18.248
LM = √((8-2)^2 +(-3-2)^2) = √(36 +25) = √61 ≈ 7.810
So, the perimeter is ...
P = 6 + 10 + 18.248 +7.810 = 42.058 ≈ 42 units
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Here, it is helpful to be familiar with the 3-4-5 right triangle. It has several interesting properties, one of which is that it shows up in algebra problems a lot. Any triangle with this ratio of side lengths is also a right triangle.
In this problem, the difference in coordinates K - N = (-4-(-10), 2-(-6)) = (6, 8) which we recognize as having the ratio 3:4. We could continue with the distance formula:
NK = √(6^2 +8^2) = √100 = 10
or, we can simply recognize this will be the result based on our familiarity with this triangle.
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An alternate approach to this problem will also work. You can estimate the length of the perimeter.
The distance between two points is more than the maximum difference of their coordinates and less than the sum of differences of their coordinates. For example, the distance between points N and K will be more than 8 and less than 8+6=14.
If you need to refine this very crude estimate further, you can add 40% of the smallest difference to the largest difference. In this case, that would be ...
8 + 0.40·6 = 10.4 . . . . . we already know the length is actually 10, so we see this estimate is within 4% of the real length.
For the coordinates in this problem, we can see that the perimeter will be more than the sum of the longest coordinate differences: 6+6+18+8 = 38. This is an important fact, because it eliminates all of the answer choices except 42. If there were any remaining ambiguity as to the answer, we could refine our estimate by adding 40% of the sum of the shortest coordinate differences: 0.40·(0+5+3+6) = 5.6. That would bring our estimate to 38+5.6 = 43.6, within 4% of the actual value of the perimeter.
