Geometry: Traingles

Yaj wrote:Need some help -

If the sides of an isosceles triangle are a-4, a-6, and 2a-20, respectively, which of the following could be the perimeter of the triangle?
I. 26
II. 32
III. 34
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

[spoiler]OA: E[/spoiler]

Thanks !!

If we have an isosceles triangle, then 2 of the 3 sides must have equal length. So, let's see what happens with each possible case.

Case 1: the sides with lengths a-4 and 2a-20 are equal
In other words, a - 4 = 2a - 20
Solve, to get: a = 16
Plug in 16 for a to get the 3 lengths: 12, 10 and 12
So, in this case the perimeter = 12 + 10 + 12 = 34

Case 2: the sides with lengths a-6 and 2a-20 are equal
In other words, a - 6 = 2a - 20
Solve, to get: a = 14
Plug in 16 for a to get the 3 lengths: 10, 8 and 8
So, in this case the perimeter = 10 + 8 + 8 = 26

Case 3: the sides with lengths a-4 and a-6 are equal
In other words, a - 4 = a - 6
IMPOSSIBLE!!
So, case 3 cannot occur.

Answer: E

Cheers,
Brent