If the lengths of two sides of a triangle are 12 and 15, which of the following must be true of p, the perimeter of the triangle?
A. 3<p<27
B. 12<p<42
C. 12<p<54
D. 30<p<54
E. 3<p<54
I honestly don't know what this question is doing in my summer hw and i need help.
I'm dumb, I know
To solve this question, we need to use the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
In this case, let's assume that the third side of the triangle has length x. So the three sides of the triangle would be 12, 15, and x.
Now, let's try to determine the possible range of values for x.
According to the triangle inequality theorem, for the side lengths of 12, 15, and x, we have:
12 + 15 > x (Because the sum of the two shortest sides must be greater than the longest side)
Simplifying the inequality, we have:
27 > x
This means that x must be less than 27.
To find the perimeter of the triangle, we need to add up all three sides:
p = 12 + 15 + x
Since we want to determine the range of possible values for p, we can substitute the maximum value for x, which is 26.99.
p = 12 + 15 + 26.99
p ≈ 53.99
So, the perimeter of the triangle is approximately 53.99.
Now, let's analyze each option to see which one is true.
A. 3 < p < 27: This is not necessarily true because p can be greater than 27.
B. 12 < p < 42: This is not necessarily true because p can be greater than 42.
C. 12 < p < 54: This is not true because p is not less than 12.
D. 30 < p < 54: This is not true because p can be less than 30.
E. 3 < p < 54: This is true because we know that p is greater than 3 and less than 54.
Therefore, the correct answer is (E) 3 < p < 54.