Question

A carpenter proposes to make 4 triangular shelves of the following dimensions:

Shelf 1: 6 inches, 12 inches, 17 inches
Shelf 2: 10 inches, 14 inches, 26 inches
Shelf 3: 11 inches, 20 inches, 32 inches
Shelf 4: 8 inches, 20 inches, 23 inches

Which of the above shelf dimensions are not possible?

Shelf 1 and Shelf 4
Shelf 2 and Shelf 4
Shelf 2 and Shelf 3
Shelf 1 and Shelf 3

Answer 1

The solution of this problem is a direct application of the following theorem:

Triangle Inequality Theorem

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

I think of it in an equivalent way:
One side of a triangle is less than the sum of the other two.

Answer 2

Shelf 1: 6 inches, 12 inches, 17 inches
Starting with Shelf One's dimensions, there are three checks to make>
Is 6 < 12 + 17 YES
Is 12 < 6 + 17 YES
Is 17 < 6 + 12 YES
Conclusion: A triangles shelf can be made with these dimensions.

Answer 3

Shelf 2: 10 inches, 14 inches, 26 inches
Is 10 < 14 + 26 YES
Is 14 < 10 + 26 ? YES
Is 26 < 10 + 14 NO
Conclusion: A triangular shelf cannot be made with these dimensions.

Answer 4

Shelf 3: 11 inches, 20 inches, 32 inches
Is 11 < 20 + 32 YES
Is 20 < 11 + 32 YES
Is 32 < 11 + 20 NO
Conclusion: No triangular shelf can be made with these dimensions.

Answer 5

Looking at the options, there is no need to check shelf 4.

Which of the above shelf dimensions are not possible?
---> Shelf 2 and Shelf 3