A triangle has two sides of length 8 and 12. What is the largest possible whole-number length for the third side?

To find the largest possible whole-number length for the third side of a triangle with two sides measuring 8 and 12, we can use the **triangle inequality theorem**. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the third side be denoted by \( x \). The triangle inequality gives us three conditions:

1. \( 8 + 12 > x \) → \( x < 20 \)
2. \( 8 + x > 12 \) → \( x > 4 \)
3. \( 12 + x > 8 \) → \( x > -4 \) (which is always true since \( x \) is positive)

Thus, \( x \) must satisfy:

\[
4 < x < 20
\]

The largest possible whole-number length for the third side is \( \boxed{19} \).

Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. What is the smallest possible value for the third side of the triangle?
2. How does the triangle inequality theorem apply to different types of triangles?
3. If the triangle were a right triangle, how would the third side change?
4. How would the triangle’s area be affected by changing the third side?
5. What happens to the angles in a triangle as you change the length of one side?

**Tip:** Always remember that the sum of any two sides of a triangle must be greater than the third side to form a valid triangle.

A triangle has two sides of length 8 and 12. What is the largest possible whole-number length for the third side?

Explore how to find the largest possible whole-number length for the third side of a triangle with sides 8 and 12 using the Triangle Inequality Theorem. This geometry problem demonstrates how to apply key theorems and is ideal for students in Grades 7-9.

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