Find which of the following can be the dimensions of the sides of a triangle.
(a). 5 cm, 9 cm, 13 cm
(b) 6 cm, 8 cm, 15 cm
(c) 12 cm, 10 cm, 9 cm
(d) 3 cm, 7 cm, 3 cm.

Answer

To determine which of the given sets of dimensions can form a triangle, we will use the triangle inequality theorem. According to this theorem, for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following conditions must be satisfied:

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

Now, let's analyze each option step by step.

### Step 1: Check the dimensions 5 cm, 9 cm, and 13 cm
- Let \( a = 5 \), \( b = 9 \), \( c = 13 \).
- Check \( a + b > c \):
\( 5 + 9 = 14 > 13 \) (True)
- Check \( a + c > b \):
\( 5 + 13 = 18 > 9 \) (True)
- Check \( b + c > a \):
\( 9 + 13 = 22 > 5 \) (True)

**Conclusion**: This set can form a triangle.

### Step 2: Check the dimensions 6 cm, 8 cm, and 15 cm
- Let \( a = 6 \), \( b = 8 \), \( c = 15 \).
- Check \( a + b > c \):
\( 6 + 8 = 14 < 15 \) (False)

Since the first condition fails, we do not need to check the others.

**Conclusion**: This set cannot form a triangle.

### Step 3: Check the dimensions 12 cm, 10 cm, and 9 cm
- Let \( a = 12 \), \( b = 10 \), \( c = 9 \).
- Check \( a + b > c \):
\( 12 + 10 = 22 > 9 \) (True)
- Check \( a + c > b \):
\( 12 + 9 = 21 > 10 \) (True)
- Check \( b + c > a \):
\( 10 + 9 = 19 > 12 \) (True)

**Conclusion**: This set can form a triangle.

### Step 4: Check the dimensions 3 cm, 7 cm, and 3 cm
- Let \( a = 3 \), \( b = 7 \), \( c = 3 \).
- Check \( a + b > c \):
\( 3 + 7 = 10 > 3 \) (True)
- Check \( a + c > b \):
\( 3 + 3 = 6 < 7 \) (False)

Since the second condition fails, we do not need to check the others.

**Conclusion**: This set cannot form a triangle.

### Final Results:
- (a) 5 cm, 9 cm, 13 cm: **Can form a triangle**
- (b) 6 cm, 8 cm, 15 cm: **Cannot form a triangle**
- (c) 12 cm, 10 cm, 9 cm: **Can form a triangle**
- (d) 3 cm, 7 cm, 3 cm: **Cannot form a triangle**

### Summary:
- **A**: True
- **B**: False
- **C**: True
- **D**: False

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