Discussion
One side of a triangle is 5 cm and the other side is 10 cm and its perimeter is P cm, where P is an integer. The least and the greatest possible values of P are respectively
Answer: C
The least and greatest values of P are 21 and 29.
By the triangle inequality, the third side c must be greater than the difference of the other two sides and less than their sum:
\[ 10 - 5 \lt c \lt 10 + 5, \quad 5 \lt c \lt 15 \]
Since P = 15 + c is an integer, c is an integer from 6 to 14. So P runs from 15 + 6 = 21 to 15 + 14 = 29.
19 and 20 would need a third side of 4 or 5, which cannot close the triangle, and 28 or 27 as the greatest value stops short of c = 14, which is allowed.
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