A cash register in a certain clothing store is the same distance from two dressing rooms in the store.If the distance between the two dressing rooms is 16 feet, which of the following could be the distance between the cash register and either dressing room?
I 6 feet
II 12 feet
III 24 feet
I only
II only
III only
I and II
II and III
Let A = the cash register, B = one of the dressing rooms, and C = the other dressing room.
The three points A, B, and C form ∆ABC.
Since the cash register is the same distance from each dressing room, AB = AC.
Since the distance between the dressing rooms is 16 feet, BC = 16.
Thus, ∆ABC is an isosceles triangle, where AB=AC and BC = 16.
The THIRD SIDE of a triangle must be LESS THAN THE SUM of the lengths of the other two sides.
Thus:
BC < AB+AC
16 < AB+AC
AB + AC > 16.
Since AB=AC, we get:
AB + AB > 16
2(AB) > 16
AB > 8.
Of the three options for AB -- 6, 12, and 24 -- only II and III are possible.
The correct answer is
E.
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