Question

1

Which one of the following sets of
numbers could represent the lengths of
the sides of a triangle?
A 8
{8, 11, 19}
B {19, 16, 20}
C {3, 4, 8}
D {11, 5, 5

Answer 1

B. {16, 19, 20}

Explanation:

The triangle inequality requires for any sides a, b, c you must have ...

a + b > c

b + c > a

c + a > b

The net result of those requirements are ...

• the sum of the two shortest sides must be greater than the longest side
• the length of the third side lies between the difference and sum of the other two sides

__

If we look at the offered side length choices, we see ...

A: 8+11 = 19 . . . not > 19; not a triangle

B: 16+19 = 35 > 20; could be a triangle

C: 3+4 = 7 . . . not > 8; not a triangle

D: 5+5 = 10 . . . not > 11; not a triangle

The side lengths {16, 19, 20} could represent the sides of a triangle.

_____

Additional comment

The version of triangle inequality shown above ensures that a triangle will have non-zero area.

The alternative version of the triangle inequality uses ≥ instead of >. Triangles where a+b=c will look like a line segment--they will have zero area. Many authors disallow this case. (If it were allowed, then {8, 11, 19} would also be a "triangle.")