High school geometry: Triangle Inequality test questions

📚 Quick Study Guide

• 📐 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 side lengths of a triangle, then the following inequalities must be true: $a + b > c$, $a + c > b$, and $b + c > a$.
• 💡 To check if three given side lengths can form a triangle, simply add the two smallest lengths and see if their sum is greater than the largest length.
• 🧮 If the sum of the two smallest sides is equal to or less than the largest side, then a triangle cannot be formed.
• 📏 This theorem helps determine the validity of triangle construction based on given side lengths.

Practice Quiz

1. Which of the following sets of side lengths can form a triangle?
   1. 3, 4, 8
   2. 5, 6, 7
   3. 2, 2, 4
   4. 1, 5, 6
2. If two sides of a triangle are 5 and 8, what is a possible length for the third side?
   1. 2
   2. 3
   3. 12
   4. 13
3. Which of the following sets of side lengths CANNOT form a triangle?
   1. 6, 8, 10
   2. 4, 5, 6
   3. 1, 2, 5
   4. 7, 9, 11
4. Two sides of a triangle are 9 and 15. Which value could be the length of the third side?
   1. 5
   2. 6
   3. 24
   4. 25
5. Which set of lengths could be the sides of a triangle?
   1. 1, 1, 3
   2. 4, 5, 9
   3. 5, 12, 13
   4. 2, 3, 6
6. The lengths of two sides of a triangle are 7 and 10. What is the range of possible values for the third side, $x$?
   1. $x < 3$
   2. $x > 17$
   3. $3 < x < 17$
   4. $x = 17$
7. Which of the following side lengths would NOT form a triangle with sides of length 4 and 6?
   1. 3
   2. 5
   3. 9
   4. 10