Question
Which of the following sets of numbers could not represent the three sides of a triangle? Answer \( \{5,9,13\} \) \( \{9,21,29\} \) \( \{10,20,31\} \) \( \{13,27,39\} \) Submit Answer
Ask by Ramirez Rose. in the United States
Jan 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The set \(\{10, 20, 31\}\) cannot represent the sides of a triangle.
Solution
To determine which set of numbers **cannot** represent the sides of a triangle, we use the **triangle inequality theorem**. This theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Let's evaluate each set:
1. **\(\{5, 9, 13\}\)**
- \(5 + 9 = 14 > 13\) ✓
- \(5 + 13 = 18 > 9\) ✓
- \(9 + 13 = 22 > 5\) ✓
**Valid triangle**
2. **\(\{9, 21, 29\}\)**
- \(9 + 21 = 30 > 29\) ✓
- \(9 + 29 = 38 > 21\) ✓
- \(21 + 29 = 50 > 9\) ✓
**Valid triangle**
3. **\(\{10, 20, 31\}\)**
- \(10 + 20 = 30 \nleq 31\) ✗
**Invalid triangle**
4. **\(\{13, 27, 39\}\)**
- \(13 + 27 = 40 > 39\) ✓
- \(13 + 39 = 52 > 27\) ✓
- \(27 + 39 = 66 > 13\) ✓
**Valid triangle**
**Conclusion:** The set \(\{10, 20, 31\}\) **cannot** represent the sides of a triangle.
**Answer:** \( \{10,20,31\} \)
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Simplify this solution The Deep Dive
To determine which sets of numbers cannot represent the sides of a triangle, 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 the set \( \{10,20,31\} \): - \( 10 + 20 = 30 \) which is not greater than \( 31 \). Thus, this set cannot represent a triangle! Keep an eye out for those tricky combinations when you’re analyzing side lengths; they can sneak up on you and trip you up like a classic triangle inequality misstep!
