Question

Two sides of a triangle are 5 centimeters and 6 centimeters. What is the range of possible lengths for the third side? Explain your reasoning using complete sentences.

Answer 1

According to the Triangle Inequality Theorem, the range of possible lengths for the third side of the triangle, x, is 1 < x < 11.

Explanation:

To determine the range of possible lengths for the third side of the triangle, we need to use the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

If a, b, and c are the lengths of the sides of a triangle, then:

• a + b > c
• a + c > b
• b + c > a

We have been told that two sides of the triangle are 5 cm and 6 cm.

Let "x" be the length of the third of the triangle.

Using the Triangle Inequality Theorem, we can write the following inequalities:

• 5 + 6 > x
• 6 + x > 5
• 5 + x > 6

Simplify the inequalities:

• 11 > x
• x > - 1
• x > 1

The first inequality tells us that x should be less than 11 cm.

The second inequality tells us that x should be greater than zero (since length cannot be negative).

The third inequality tells us that the x should be greater than 1 cm.

Therefore, the range of possible lengths for the third side is 1 < x < 11.