8. EXPLAIN if the three lengths can form a triangle. $$ 3,4,7 $$ Yes because ... No because ...

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To determine if the 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. $$ b + c > a $$

Let's denote the lengths as follows:
- $$ a = 3 $$
- $$ b = 4 $$
- $$ c = 7 $$

Now, we will check each of the triangle inequality conditions:

1. Check $$ a + b > c $$:
   $$
   3 + 4 > 7 \quad \Rightarrow \quad 7 > 7 \quad \text{(not true)}
   $$

2. Check $$ a + c > b $$:
   $$
   3 + 7 > 4 \quad \Rightarrow \quad 10 > 4 \quad \text{(true)}
   $$

3. Check $$ b + c > a $$:
   $$
   4 + 7 > 3 \quad \Rightarrow \quad 11 > 3 \quad \text{(true)}
   $$

Since the first condition $$ 3 + 4 > 7 $$ is not satisfied, the three lengths cannot form a triangle.

Thus, the answer is:
**No, because the sum of the two shorter lengths (3 and 4) is not greater than the longest length (7).**
No, because the sum of the two shorter lengths (3 and 4) is not greater than the longest length (7).

8. EXPLAIN if the three lengths can form a triangle. \[ 3,4,7 \] Yes because ... No because ...

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