sanju09 wrote:Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?
I. 3
II. 9 √3
III. 13.5

Length of AB must be within the difference and sum of the lengths of AC and BC, i.e. within 3 and 15 (excluding 3 and 15). Only such option is 13.5.

Hence the correct answer is C.

sanju09 wrote:Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?
I. 3
II. 9 √3
III. 13.5
(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

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

So, based on the rule, we know that: 9-6 < third side < 9+6
In other words, 3 < third side < 15

Now check the 3 statements:
I. 3 - NOT within the range of 3 < third side < 15

II. 9√3 - all students should memorize that √3 ≈ 1.7 (along with √2 ≈ 1.4 and √5 ≈ 2.2)
So 9√3 ≈ 9(1.7) ≈ 15.3. This is NOT within the range of 3 < third side < 15

III. 13.5 WORKS! 13.5 IS within the range of 3 < third side < 15

Answer: C

sanju09 wrote:Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

This problem is testing us on the triangle inequality theorem in which the sum of the lengths any two sides of a triangle MUST be greater than that of the third side. Let's test each Roman numeral to determine which could be a possible side length of AB. Keep in mind that AC = 6 and BC = 9.

I. 3

Since 6 + 3 IS NOT greater than 9, 3 cannot be the length of side AB.

II. 9√3

9√3 is roughly equal to 9 x 1.7 = 15.3, and since 6 + 9 IS NOT greater than 15.3, 9√3 cannot be the length of side AB.

III. 13.5

Since 6 + 13.5 IS greater than 9, 13.5 + 9 IS greater than 6, and 9 + 6 IS greater than 13.5; thus, 13.5 could be the length of side AB.

Answer: C