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In ΔABC above, which of the following must be true?

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In ΔABC above, which of the following must be true?  [#permalink]

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New post 27 Dec 2015, 10:45
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A
B
C
D
E

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  35% (medium)

Question Stats:

69% (01:27) correct 31% (01:35) wrong based on 132 sessions

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Re: In ΔABC above, which of the following must be true?  [#permalink]

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New post 27 Dec 2015, 21:59
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Bunuel wrote:
Image
In ΔABC above, which of the following must be true?

I. x > 50
II. AC < 10
III. AB > 10



Attachment:
2015-12-27_2143.png


I. x > 50
3x-15=180
x=65
True

II. AC < 10
The side opposite of angle x is 10, so the side opposite of angle x-15 must be less than 10
True

III. AB > 10
The side opposite of angle x is 10, so the side opposite of angle x-15 must be less than 10
False

C. I and II only
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Re: In ΔABC above, which of the following must be true?  [#permalink]

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New post 28 Dec 2015, 23:08
Hi All,

In Roman Numeral questions, it helps to pay attention to how the answer choices are designed - there will almost always be some type of built-in 'logic shortcut' that will allow you to avoid some of the work involved. This question is built around a subtle rule regarding triangle side lengths and the angles that are across from them:

The BIGGER the angle, the BIGGER the side that's across from it (and similarly, the SMALLER the angle, the SMALLER the side that's across from it).

From the diagram, we know that we're dealing with an ISOSCELES triangle (angles A and C are the same), so we know that the sides that are across from those two angles are the SAME (they're both 10). The third angle (angle B) is SMALLER than the other two angles, so the side across from angle B is SMALLER than the other two sides (it's LESS than 10).

With this knowledge, we know that Roman Numeral 2 is TRUE and Roman Numeral 3 is NOT TRUE. There's only one answer that matches (and we don't even have to deal with Roman Numeral 1).

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Re: In ΔABC above, which of the following must be true?  [#permalink]

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New post 27 Jan 2019, 02:12
Bunuel wrote:
Image
In ΔABC above, which of the following must be true?

I. x > 50
II. AC < 10
III. AB > 10

A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III

Attachment:
2015-12-27_2143.png


My silly question ,
we know the Triangle equality theorem which states that the third side will be greater than the difference of the other two sides and less than the sum of the other two sides. Based on this rule a triangle having two sides as 10 and 10 should allow a third side between 1 and 19. So third side being 15 or 16 etc is a possibility according to this rule.

So why in this question this rule is literally not followed? Or do we need to additionally check the angles to make sure , that both the triangle inequality rule and angle rule is maintained?
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Re: In ΔABC above, which of the following must be true?  [#permalink]

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New post 27 Jan 2019, 03:14
1
stne wrote:
Bunuel wrote:
Image
In ΔABC above, which of the following must be true?

I. x > 50
II. AC < 10
III. AB > 10

A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III

Attachment:
2015-12-27_2143.png


My silly question ,
we know the Triangle equality theorem which states that the third side will be greater than the difference of the other two sides and less than the sum of the other two sides. Based on this rule a triangle having two sides as 10 and 10 should allow a third side between 1 and 19. So third side being 15 or 16 etc is a possibility according to this rule.

So why in this question this rule is literally not followed? Or do we need to additionally check the angles to make sure , that both the triangle inequality rule and angle rule is maintained?


If we knew only that there is an isosceles triangle with two equal sides of 10, then the third side could be 0 < (third side) < 20. Notice that it's NOT 1 < (third side) < 19, as you've written.

But in the question at hand we know more. Usually every bit of additional information allows us to narrow the answer. So, here knowing that the third side is opposite the smallest angle in the triangle allows us to conclude that the third side must be the shortest among the three, thus (third side) < 10.
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Re: In ΔABC above, which of the following must be true?  [#permalink]

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New post 27 Jan 2019, 04:01
Bunuel wrote:
stne wrote:
Bunuel wrote:
Image
In ΔABC above, which of the following must be true?

I. x > 50
II. AC < 10
III. AB > 10

A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III

Attachment:
2015-12-27_2143.png


My silly question ,
we know the Triangle equality theorem which states that the third side will be greater than the difference of the other two sides and less than the sum of the other two sides. Based on this rule a triangle having two sides as 10 and 10 should allow a third side between 1 and 19. So third side being 15 or 16 etc is a possibility according to this rule.

So why in this question this rule is literally not followed? Or do we need to additionally check the angles to make sure , that both the triangle inequality rule and angle rule is maintained?


If we knew only that there is an isosceles triangle with two equal sides of 10, then the third side could be 0 < (third side) < 20. Notice that it's NOT 1 < (third side) < 19, as you've written.

But in the question at hand we know more. Usually every bit of additional information allows us to narrow the answer. So, here knowing that the third side is opposite the smallest angle in the triangle allows us to conclude that the third side must be the shortest among the three, thus (third side) < 10.


Thank you, initially convinced myself that perhaps this rule was not applicable for isosceles triangle , though couldn't find anything anywhere about such an exception.Your reply really helped to clear this up.
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Re: In ΔABC above, which of the following must be true?   [#permalink] 27 Jan 2019, 04:01
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