Question

If two sides of a triangle are 6 and 16, what is the range of the possible lengths of the third side? Explain your answer thoroughly.

Answer 1

The possible length of the third side of a triangle with sides 6 and 16 must be greater than 10 and less than 22 for it to form a valid triangle, according to the triangle inequality theorem.

Step-by-step explanation:

To determine the range of the possible lengths for the third side of a triangle when two sides are known, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side for all three sides to form a triangle.

Let's denote the unknown side's length as x. Since we're given two sides of lengths 6 and 16, we can set up the following inequalities:

• x + 6 > 16
• x + 16 > 6
• 6 + 16 > x

By solving these inequalities, we find:

• x > 10
• x > -10 (which is always true for the length of a side)
• x < 22

Therefore, the range of the possible lengths for the third side is from >10 to <22, not inclusive of 10 and 22 because the third side must be strictly longer than the difference and strictly shorter than the sum of the other two sides.

Answer 2

In a triangle the sum of two sides is always greater than the third side.

16-6 < x < 16+6
10 < x < 22 ← answer