BTGmoderatorDC wrote:If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?
(A) one
(B) two
(C) three
(D) four
(E) five
OA A
Source: Official Guide
Recall the triangle inequality, which states that in order for three values to be the lengths of the sides of a triangle, the sum of the two smaller values must be greater than the third (the largest value).
Since k is an integer between 2 and 7, k can be 3, 4, 5 or 6.
If k = 3, {2, 3, 7} can't be the lengths of the sides of a triangle since 2 + 3 = 5 is not greater than 7.
If k = 4, {2, 4, 7} can't be the lengths of the sides of a triangle since 2 + 4 = 6 is not greater than 7.
If k = 5, {2, 5, 7} can't be the lengths of the sides of a triangle since 2 + 3 = 7 is not greater than 7.
If k = 6, {2, 6, 7} CAN be the lengths of the sides of a triangle since 2 + 6 = 8 is greater than 7.
Answer: A
