Problem Solving

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

IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
DIFFERENCE between A and B < length of third side < SUM of A and B

We're told that the two KNOWN sides have lengths 2 and 7
So, we can write: (7 - 2) < k < (7 + 2)
Simplify: 5 < k < 9
Since k is an INTEGER, k can equal 6, 7 or 8

However, we're also told that 2 < k < 7, so 6 is the only value that k can have.

Answer: A

Cheers,
Brent

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