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Geometry: Traingles

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by Yaj » Thu Sep 12, 2013 2:45 am
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 !!
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by ganeshrkamath » Thu Sep 12, 2013 2:58 am
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 !!
Perimeter = a-4 + a-6 + 2a-20
= 4a - 30

Assuming a to be an integer,
the (answer + 30) should be a multiple of 4.

26 + 30 = 56 ---- YES
32 + 30 = 62 ---- NO
34 + 30 = 64 ---- YES

Choose E

However, without making the above-mentioned assumption, I don't know how to arrive at the answer.

Cheers
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by Brent@GMATPrepNow » Thu Sep 12, 2013 5:58 am
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
Brent Hanneson - Creator of GMATPrepNow.com
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