Question

How many triangles exist with the given side lengths?

3cm,5cm,9cm

A.
Exactly one unique triangle exists with the given side lengths.

B.
No triangle exists with the given side lengths.

C.
More than one unique triangle exists with the given side lengths.

Answer 1

To determine whether a triangle can exist with the given side lengths of 3 cm, 5 cm, and 9 cm, we need to apply the triangle inequality theorem. According to this theorem, 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 side lengths as \( a = 3 \), \( b = 5 \), and \( c = 9 \). Now we will check the triangle inequalities:

1. \( 3 + 5 > 9 \) ⟹ \( 8 > 9 \) (FALSE)
2. \( 3 + 9 > 5 \) ⟹ \( 12 > 5 \) (TRUE)
3. \( 5 + 9 > 3 \) ⟹ \( 14 > 3 \) (TRUE)

Since the first condition \( 3 + 5 > 9 \) is false, the three lengths cannot form a triangle.

Therefore, the answer is:

**B. No triangle exists with the given side lengths.**