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# Is triangle ABC isosceles?

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Re: Is triangle ABC isosceles? [#permalink]
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Is triangle ABC isosceles?

(1) X does not equal Y --> angles at the side BC are not equal --> , but the third side, BC, can be equal to either of them. Not sufficient.

Tells us nothing. X ≠ Y but the third angle could equal x or y in which case the triangle would be isosceles or the third angle would = w in which case the triangle would not be isosceles. Insufficient.

(2) AB/BC=2

The length of AB is twice the length of BC but we don't know anything about side AC. Insufficient.

1+2) The only way for this triangle to be isosceles is if a.) AC = BC or b.) AB = AC. Neither of these outcomes are possible. If AC = AB then a triangle wouldn't be possible. In any triangle, the sum of two legs must be greater than the third. If AC = BC then the formula for the triangle would be 2BC = BC + BC which isn't possible. We also know that AC ≠ AB because the angles opposite them, x and y respectively, do not equal one another. Thus, AC cannot = AB and AC is a different length than AB. This triangle CANNOT be isosceles. SUFFICIENT.

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Re: Is triangle ABC isosceles? [#permalink]
Bunuel Statement 2 says AB=2BC. Lets say BC=x then, AB=2x.
Thus, the third side has to be less than x and greater than 3x. So the triangle can't be isosceles.
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Re: Is triangle ABC isosceles? [#permalink]
NehaBhargava wrote:
Bunuel Statement 2 says AB=2BC. Lets say BC=x then, AB=2x.
Thus, the third side has to be less than x and greater than 3x. So the triangle can't be isosceles.

Why not? The third side can have the length of 2x. For example, you can have an isosceles triangle with sides 1, 2, and 2.
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Re: Is triangle ABC isosceles? [#permalink]
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NehaBhargava wrote:
Bunuel Statement 2 says AB=2BC. Lets say BC=x then, AB=2x.
Thus, the third side has to be less than x and greater than 3x. So the triangle can't be isosceles.

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

Thus, if the length of two sides are x and 2x, then:

(2x - x) < (third side) < (2x + x)

x < (third side) < 3x.
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Re: Is triangle ABC isosceles? [#permalink]
NehaBhargava wrote:
Bunuel Statement 2 says AB=2BC. Lets say BC=x then, AB=2x.
Thus, the third side has to be less than x and greater than 3x. So the triangle can't be isosceles.

Hi NehaBhargava

If BC = x and AB = 2x, the third side has to be LESS than 3x (the sum of two sides is greater than the 3rd side in a triangle)
The third side can still be equal to 2x, so this statement is not sufficient.

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Re: Is triangle ABC isosceles? [#permalink]
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C.

Very good question.

(1) not sufficient
because x <> y, but x could be equal to angle A (isos triangle). So triangle could be isos or not

(2) not sufficient because :
let AB = 4 , and BC = 2. This satisfies the condition in statement 2.

Now, if AC = 4 , triangle is isos. (It's also possible because it satisfies triangles property that sum of two sides is greater than the third side).

But, if AC = 5, triangle is NOT isos, but this is a possible scenario.

Combine both statements, it's sufficient because for triangle to be an isos, AC must be either 4 or 2.
But, since x <> y, AC cannot be 4. Also, AC cannot be 2 because then it goes against the triangle property that sum of two sides is greater than the third side.

This by combining we see that the triangle can NEVER be isos. Sufficient.

Regards,
Saakhi
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Re: Is triangle ABC isosceles? [#permalink]
the key point in this question is the ratio 2:1 (ab/bc=2). If there is no this special ratio-2:1, the answer will be E. They are checking whether students know the conception: the sum of the two sides is always greater than the third side. In this question, if AC=BC, it violates this theorem.
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Re: Is triangle ABC isosceles? [#permalink]
Is triangle ABC isosceles?

(1) X does not equal Y
(2) AB/BC=2

From the question stem we can deduce a few possibilities for Triangle ABC to be isosceles:-
a) x=y
b) x= Angle BAC
c) y= Angle BAC

Statement (1) X does not equal Y

Even if X does not equal Y, either X or Y can be equal to Angle BAC. Or neither X nor Y will be equal to Angle BAC.
Hence we have two possibilities therefore insufficient.

Statement (2) AB/BC=2

AB = 2BC

Say BC = m
Then, AB = 2m

Take examples: AB = 8
BC = 4

In a triangle sum of two sides is greater than the third side,
Hence in triangle ABC with sides 8,4 and AC

4<AC<12

Therefore, the value of AC can be any number from 4.1 to 11.9

For triangle ABC to be isosceles AC has to take the value of 8, which would make AC= AB=8.
But AC can take any value, for eg, 5,6,7,8,9..11

Hence statement two is also insufficient.

(1) and (2)

From statement (2) we know that BC can not be equal to AC
From statement (1) we have AB is not equal to AC

Hence ABC is not an isosceles triangle