Question

Triangle A B C is shown. Side A C has a length of 27. Side C B has a length of 54.

Based on the diagram, which expresses all possible lengths of segment AB?

AB = 25
27 < AB < 81
AB = 85
AB< 27 or AB > 81

Answer 1

Answer:27 < AB < 81

Explanation:

Answer 2

27 < AB < 81

Explanation:

in a triangle the sum of any 2 sides must be larger than the third side.

that must be true for every pair of sides you pick.

simply because, if that is not fulfilled, the third side would be too long, and the other 2 sides cannot connect to the endpoints of the third side.

so,

AC + CB = 81 must be longer than AB. otherwise the triangle cannot "close".

AC + AB = 27 + AB must be longer than CB = 54.

so, AB must be longer than 27.

CB + AB = 54 + AB must be longer than AC = 27. but that is always true, as CB alone is already larger.

so only the other 2 conditions need to be true.

hence the answer.