10. CBQ - The Hiking route

Question:

4 m.

Three campsites: Alpha ($A$), Beta ($B$), and Gamma ($C$) are connected by straight walking trails. The trail distance between Alpha and Beta is $6 \ \text{km}$, and between Beta and Gamma is $14 \ \text{km}$. A group of hikers wants to plan a third trail connecting Gamma directly back to Alpha.

1. What is the minimum possible distance (strictly greater than) for the direct trail between Gamma and Alpha?

$20 km$
$8 km$
$22 km$
$2 km$

2. What is the maximum possible distance (strictly less than) for the direct trail between Gamma and Alpha?

$20 km$
$2 km$
$22 km$
$8 km$

3. If the forest department attempts to build a straight trail of $22 \ \text{km}$ between Gamma and Alpha, could the three campsites form a triangular route?

Yes, because any $3$ lengths form a triangle.
Yes, because $6 + 14 < 22$, which satisfies the triangle inequality.
No, because all three sides must be equal.
No, because $6 + 14 < 22$, which does not satisfy the triangle inequality.

4. Which rule guarantees whether a triangle can be formed given three side lengths \(

a

, \(

b

, and \(

c

$a - b > c$
$a + b = c$
$a + b > c$, $a + c > b$ and $b + c > a$
$a + b < c$

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