basically the cash register location alongwith the two dressing rooms with make an isosceles triangle

Property: For a triangle to exist Sum of the two shorter sides must be greater than the longest side

I. 6 feet --- i.e. 6+6 is NOT greater than 16 hence INCORRECT
II. 12 feet --- i.e. 12+12 is greater than 16 hence CORRECT
III. 24 feet --- i.e. 16+24 is greater than 24 hence CORRECT

Answer: option E
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We see that we have an isosceles triangle with its base equal to 16 feet. The other two sides must be equal. We can use the triangle inequality theorem to determine the possible lengths of those two equal sides. The theorem states that the sum of the lengths of the two shortest sides is greater than the length of the longest side.

I. 6 feet

The resulting triangle would have sides of 6, 6, and 16. The two other sides cannot be 6 since 6 + 6 is not greater than 16.

II. 12 feet

The resulting triangle would have sides of 12, 12, and 16. The two other sides can be 12 since 12 + 12 is greater than 16.

III. 24 feet

The resulting triangle would have sides of 24, 24, and 16. The two other sides can be 24 feet since 16 + 24 is greater than 24. (Note in this case that the two shortest sides are 16 and 24.)

Answer: E
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can we also find out the range of the 2 sides and see if they fall between the sides.

Why can't the placement be in a straight line? Is there something in the statement that makes it a triangle?

How can both dressing rooms be of equal distance from register if they are in vertical line?

the second dressing room in line will be much farther than the original one.

Could this also be solved using the Triangle property of : The length of any side of a triangle MUST BE LARGER than the positive difference of the other 2 sides, but SMALLER than the sum of the other 2 sides?

I ) 6-6<16< 6+6 => 0<16<12 (NOT POSSIBLE)
II) 12-12 <16<12+12 => 0<16<24 (POSSIBLE)
III) 24-24 < 16 < 24+24 => 0<16>48 (POSSIBLE)

Is my understanding of this property correct?

Hi Bunuel, writing to share that this is a GMAT PREP question.

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Thank you. Added the tag.
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If both dressing rooms and the register are in a straight line, then the only way the register is equidistant from each room is when it is in the mid point of the line which is 16 ft/2 = 8ft. There would be only a single value 8 ft for the answer choice, which is not the case in this question which has options 6 ft, 12 ft and 24 ft.


Another way to think of this scenario is that the line discussed above [refer case I in diagram below] is also a form of triangle with angles 0 degree. Hence using the properties of the triangle: sum of 2 sides is greater than the other, the min value of each of the other sides of the triangle is 8 ft (since 8ft + 8 ft = 16 ft, sum of two side is greater than or equal to the third side)

Drawing a Straight Line from Dressing Room 1 (Point A) to Dressing Room 2 (Point B) = 16 feet

For the Cash Register to be Equidistant from each Room it must lie on the Vertical Line that is Perpendicular to this 16 foot line connecting the 2 dressing Rooms and Bisect it -----> Call this Point R


Extreme Case: All 3 Points lie on the Same Line: AR = RB = 8 feet - and the Cash Register is 8 feet away from each Room


As the Cash Register moves up the Vertical Line starting at Point R to maintain the Equidistance from each Changing Room, an Isosceles Triangle will start to form in which Sides AR and BR will be the Equal Sides.

The Non-Equal Side will be the Straight Horizontal Line from Dressing Room to Dressing Room = AB = 16 ft

Given the Isosceles Triangle ARB with NON-Equal Side of 16 ft, the Distance from the Register to each Room (the Equal Sides) can be what length?

Rule: A triangle can only be formed if the SUM of the Lengths of the 2 Shorter Sides is GREATER than the Length of the Longest Side


I. 6

If the Register is 6 feet away from each dressing room, this would form an Isosceles Triangle with Sides of: 6 - 6 - 16

However: 6 + 6 is NOT greater than 16 feet, failing the Triangle Formation Theorem

I is eliminated


II. 12

Triangle ARB would have side Lengths 12 - 12 - 16

12 + 12 > 16 --------> Valid Triangle. II is possible



III. 24

16 + 24 > 24 --------> Valid Triangle. III is possible.


II and III

(E)

How do we know that the cash register isn't 4 feet wide? I missed this question because I assumed that we couldnt rule that possibility out.