Question

Use the given conditions to determine the number of possible triangles.

first side length: 6
inches
second side length: 8
inches
third side length: 10
inches

Do the given conditions create a unique triangle, more than one triangle, or no triangle?

Responses

more than one triangle
more than one triangle

unique triangle
unique triangle

no triangle

Answer 1

To determine if the given side lengths of 6 inches, 8 inches, and 10 inches can form a triangle, we can apply the triangle inequality theorem. This theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \):

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

Let's check each condition with the given side lengths:

1. \( 6 + 8 > 10 \)
- \( 14 > 10 \) (True)

2. \( 6 + 10 > 8 \)
- \( 16 > 8 \) (True)

3. \( 8 + 10 > 6 \)
- \( 18 > 6 \) (True)

Since all three conditions are satisfied, the side lengths comply with the triangle inequality theorem, which means they can form a triangle.

Next, we will check if the triangle is unique. The given side lengths (6, 8, and 10) form a right triangle, as they satisfy the Pythagorean theorem \( a^2 + b^2 = c^2 \):

- \( 6^2 + 8^2 = 10^2 \)
- \( 36 + 64 = 100 \)
- \( 100 = 100 \) (True)

Since the side lengths define a unique configuration of a triangle (specifically, a right triangle), we conclude that there is a unique triangle with these side lengths.

Therefore, the answer is:

**unique triangle**