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

Here is the step-by-step solution to determine the possible lengths of segment AB in the given triangle ABC:

Let's break this down:

- We are given:

AC = 27

CB = 54

- We want to find: Possible lengths of AB

- To do this, we can utilize the triangle inequality theorem, which states:

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Applying this:

AC + CB > AB

27 + 54 > AB

81 > AB

- This means AB must be less than 81.

- Additionally, we know:

AB + AC > CB

AB + 27 > 54

AB > 27

- Therefore, AB must be greater than 27.

- Bringing this together:

27 < AB < 81

So the possible lengths of segment AB are:

27 < AB < 81

Therefore, the answer is option B.

By methodically applying the triangle inequality theorem, we have determined the only possible lengths for side AB given the constraints.