SOLUTION: If each of side of a triangle has a different lengths, and the lengths of two sides are 7 & 12, the perimeter of this triangle would be... A. 31 B. 26 C. 25 D. 23

Question 998615: If each of side of a triangle has a different lengths, and the lengths of two sides are 7 & 12, the perimeter of this triangle would be...
A. 31
B. 26
C. 25
D. 23

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
x = length of third side

"the lengths of two sides are 7 & 12" so

12-7 < x < 12+7
5 < x < 19

If x is the third side, then the third side is between 5 and 19 units (excluding the endpoints).

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So x is somewhere between 5 and 19. The value of x is NOT equal to 5. It is NOT equal to 19.

We also cannot have x equal to 7 or 12. Why? Because the instructions say that each side is of a different length.

Notice how if the third side was x = 7, then
x+7+12 = 7+7+12 = 26
So choice B is out because again the three sides are different lengths.

If x = 12, then
x+7+12 = 12+7+12 = 31
ruling out choice A

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So we have choice C and choice D left over.

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Focus on 5 < x < 19

Add 7 to all sides

5+7 < x+7 < 19+7
12 < x+7 < 26

Now add 12 to all sides

12 < x+7 < 26
12+12 < x+7+12 < 26+12
24 < x+7+12 < 38

The quantity "x+7+12" represents the perimeter because it's the sum of the three sides of the triangle. The perimeter is between 24 and 38 (excluding the endpoints)

As we can see in the inequality 24 < x+7+12 < 38, the perimeter is some number between 24 and 38. So it is impossible for the perimeter to be 23 because it's not in that interval. So choice D is ruled out

The only thing left is choice C and this is possible when x = 6
P = x+7+12 = 6+7+12 = 25

Final Answer: Choice C) 25
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