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Isosceles triangle

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Isosceles triangle

by ov25 » Thu Dec 09, 2010 1:21 pm
An isocls triangle has side x and y, where x >y. what is the perimeter of the triangle, in terms of x and y
1) two of the int angles > the other angle
2) perimeter > 3y
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Source: — Data Sufficiency |

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by Rahul@gurome » Thu Dec 09, 2010 7:20 pm
ov25 wrote:An isocls triangle has side x and y, where x >y. what is the perimeter of the triangle, in terms of x and y
1) two of the int angles > the other angle
2) perimeter > 3y
Given: An isosceles triangle has side x and y, where x > y. As we don't know which one of them is the equal sides, we have two possibilities for the perimeter,
  • 1. (2x + y), if x is the equal side
    1. (x + 2y), if y is the equal side
If we can identify which one of them is the equal sides, we can easily obtain the perimeter.

Statement 1: Two of the internal angles > the other angle
This means the angles opposite to the equal sides are greater than the angles opposite to the other side. Which also implies that the equal sides are greater than the third side.

Thus x is the equal side => Perimeter = (2x + y)

Sufficient.

Statement 2: Perimeter > 3y
Consider the two possible cases,
  • 1. If x is the equal side, Perimeter = (2x + y) > (2y + y) = 3y
    1. If y is the equal side, Perimeter = (x + 2y) > (y + 2y) = 3y
Thus in both cases perimeter > 3y.

Not sufficient.

The correct answer is A.

PS: In case you have any query for the explanation of the statement 2, refer to the image below. The first and third one illustrates the case of x > y and y < x respectively, while the middle one is the transitional one, i.e. the equilateral triangle.
Image
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by ov25 » Fri Dec 10, 2010 6:53 am
Rahul@gurome wrote:
Statement 1: Two of the internal angles > the other angle
This means the angles opposite to the equal sides are greater than the angles opposite to the other side. Which also implies that the equal sides are greater than the third side.
Thanks Rahul. Nice work, however, can you elaborate on this? Prolly a picture would be better

I mean there is only one angle opposite to the other side; 2x>y, x+y>y, x+y>x are the three possibilities right!. Are you saying its always that the angle opposite to greater side is the, x, is the equal angles. I am lost some where here.. appreciate your time.
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by Rahul@gurome » Fri Dec 10, 2010 11:36 am
ov25 wrote:Thanks Rahul. Nice work, however, can you elaborate on this? Prolly a picture would be better

I mean there is only one angle opposite to the other side; 2x>y, x+y>y, x+y>x are the three possibilities right!. Are you saying its always that the angle opposite to greater side is the, x, is the equal angles. I am lost some where here.. appreciate your time.
I think you have some problem understanding the statement: "Two of the internal angles > Other angle"

In an isosceles triangle, two of the three angles are same (say a) and the other one is different (say b). Then the statement says for the isosceles triangle given, two of the angles are greater than the other. This means a and a > b, as there is no other possibilities. Thus the equal angles are greater than the other. This directly implies that the sides opposite to the equal angles (the equal sides) are greater than the side opposite to the other angle (third side). This is illustrated in the image in the previous post.

This directly follows from the facts that,
  • 1. Equal sides are the sides opposite to the equal angles.
    2. If in a triangle, if angle a > angle b then side opposite to the angle a > side opposite to the angle b.
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