Question

Triangle abc has two sides that measure 5 units and 9 units. Given that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, what is an appropriate range for c, the length in units of the third side of Δ abc?

Answer 1

The range for the third side c of triangle ABC with sides of 5 units and 9 units should be greater than 4 units and less than 14 units according to the triangle inequality theorem.

Step-by-step explanation:

To determine the range for side c of triangle ABC with two sides measuring 5 units and 9 units respectively, we use the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This gives us two inequalities:

• 5 + 9 > c
• 5 + c > 9
• 9 + c > 5

Solving these inequalities, we find:

1. c < 14 (sum of the two known sides)
2. c > 4 (9 minus the shorter known side)
3. The third inequality is always true since c is positive

Therefore, the appropriate range for side c is greater than 4 units and less than 14 units.