Problem 5 Finding Possible Side Lengths

Algebra In the Solve It, you explored the possible dimensions of a triangular sandbox. Two of the sides are 5 ft and 8 ft long. What is the range of possible lengths for the third side?

Let x represent the length of the third side. Use the Triangle Inequality Theorem to write three inequalities. Then solve each inequality for x.

x + 5 > 8 x + 8 > 5 5 + 8 > x x > 3 x > − 3 x < 13 table with 2 rows and 3 columns , row1 column 1 , x plus 5 greater than 8 , column 2 x plus 8 greater than 5 , column 3 5 plus 8 greater than x , row2 column 1 , x greater than 3 , column 2 x greater than negative 3 , column 3 x less than 13 , end table

Numbers that satisfy x > 3 x greater than 3  and x > − 3 x greater than , minus 3  must be greater than 3. So, the third side must be greater than 3 ft and less than 13 ft.

✓ Got It?

5. A triangle has side lengths of 4 in. and 7 in. What is the range of possible lengths for the third side?

✓ Lesson Check

Do you know HOW?

Use Δ A B C cap delta eh b c  for Exercises 1 and 2.

1. Which side is the longest?
2. Which angle is the smallest?
3. Can a triangle have sides of lengths 4, 5, and 10? Explain.

Do you UNDERSTAND?

4. Error Analysis A friend tells you that she drew a triangle with perimeter 16 and one side of length 8. How do you know she made an error in her drawing?
5. Reasoning Is it possible to draw a right triangle with an exterior angle measuring 88? Explain your reasoning.