$$ 3 \mathrm{~cm}, 4 \mathrm{~cm} $$ and 7 cm are not the lengths of the triangle. Give the reason.

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To determine whether three lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any three lengths $$ a $$, $$ b $$, and $$ c $$ to form a triangle, the following conditions must be satisfied:

1. $$ a + b > c $$
2. $$ a + c > b $$
3. $$ b + c > a $$

In this case, the lengths are $$ a = 3 \, \text{cm} $$, $$ b = 4 \, \text{cm} $$, and $$ c = 7 \, \text{cm} $$.

Let's check each of the triangle inequality conditions:

1. $$ 3 + 4 > 7 $$
2. $$ 3 + 7 > 4 $$
3. $$ 4 + 7 > 3 $$

Now, we will evaluate these inequalities step by step.

1. **First inequality:**
   $$
   3 + 4 = 7 \quad \text{(not greater than 7)}
   $$

2. **Second inequality:**
   $$
   3 + 7 = 10 > 4 \quad \text{(true)}
   $$

3. **Third inequality:**
   $$
   4 + 7 = 11 > 3 \quad \text{(true)}
   $$

Since the first condition $$ 3 + 4 > 7 $$ is not satisfied (it is equal to 7), the lengths $$ 3 \, \text{cm} $$, $$ 4 \, \text{cm} $$, and $$ 7 \, \text{cm} $$ cannot form a triangle.

Thus, the reason is that the sum of the lengths of any two sides must be greater than the length of the third side, and in this case, it is not.
The lengths 3 cm, 4 cm, and 7 cm cannot form a triangle because the sum of the two smaller lengths (3 cm + 4 cm = 7 cm) is not greater than the largest length (7 cm).

\( 3 \mathrm{~cm}, 4 \mathrm{~cm} \) and 7 cm are not the lengths of the triangle. Give the reason.

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