Question

which set of numbers make up a triangle and which of the do not
(25,20,15)
(7,7,1)
(4,4,10)
(5,5,5)
(2,3,4)
(1,6,4)
(1,2,3)

Answer 1

To determine whether a set of three numbers can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\) (where \(a\), \(b\), and \(c\) are the lengths of the sides), the following must hold true:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

All three conditions must be satisfied for the given sides to form a triangle.

Let's evaluate each set of numbers:

1. **(25, 20, 15)**
- \(25 + 20 = 45 > 15\) ✔
- \(25 + 15 = 40 > 20\) ✔
- \(20 + 15 = 35 > 25\) ✔
- **Forms a triangle**

2. **(7, 7, 1)**
- \(7 + 7 = 14 > 1\) ✔
- \(7 + 1 = 8 > 7\) ✔
- \(7 + 1 = 8 > 7\) ✔
- **Forms a triangle**

3. **(4, 4, 10)**
- \(4 + 4 = 8 \not> 10\) ✘
- **Does not form a triangle**

4. **(5, 5, 5)**
- \(5 + 5 = 10 > 5\) ✔
- \(5 + 5 = 10 > 5\) ✔
- \(5 + 5 = 10 > 5\) ✔
- **Forms a triangle**

5. **(2, 3, 4)**
- \(2 + 3 = 5 > 4\) ✔
- \(2 + 4 = 6 > 3\) ✔
- \(3 + 4 = 7 > 2\) ✔
- **Forms a triangle**

6. **(1, 6, 4)**
- \(1 + 6 = 7 > 4\) ✔
- \(1 + 4 = 5 \not> 6\) ✘
- **Does not form a triangle**

7. **(1, 2, 3)**
- \(1 + 2 = 3 \not> 3\) ✘
- **Does not form a triangle**

### Summary:
**Forms a triangle:**
- (25, 20, 15)
- (7, 7, 1)
- (5, 5, 5)
- (2, 3, 4)

**Does not form a triangle:**
- (4, 4, 10)
- (1, 6, 4)
- (1, 2, 3)