Question
Triangle inequality theorem quick check
Apply the triangle inequality theorem to determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle
Apply the triangle inequality theorem to determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle
Answers
GPT-4o mini
The triangle inequality theorem states that for three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's denote the sides as follows:
- \( a = 4.24 \) inches
- \( b = 9.89 \) inches
- \( c = 8.75 \) inches
Now we will check the triangle inequality conditions:
1. \( a + b > c \)
\[
4.24 + 9.89 > 8.75 \\
14.13 > 8.75 \quad \text{(True)}
\]
2. \( a + c > b \)
\[
4.24 + 8.75 > 9.89 \\
13.00 > 9.89 \quad \text{(True)}
\]
3. \( b + c > a \)
\[
9.89 + 8.75 > 4.24 \\
18.64 > 4.24 \quad \text{(True)}
\]
Since all three conditions of the triangle inequality theorem are satisfied, the sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches do indeed form a triangle.
Let's denote the sides as follows:
- \( a = 4.24 \) inches
- \( b = 9.89 \) inches
- \( c = 8.75 \) inches
Now we will check the triangle inequality conditions:
1. \( a + b > c \)
\[
4.24 + 9.89 > 8.75 \\
14.13 > 8.75 \quad \text{(True)}
\]
2. \( a + c > b \)
\[
4.24 + 8.75 > 9.89 \\
13.00 > 9.89 \quad \text{(True)}
\]
3. \( b + c > a \)
\[
9.89 + 8.75 > 4.24 \\
18.64 > 4.24 \quad \text{(True)}
\]
Since all three conditions of the triangle inequality theorem are satisfied, the sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches do indeed form a triangle.