What is the possible range for the length of side LM?
Understand the Problem
The question is asking for the possible range of the length of side LM in a triangle, given the lengths of the other two sides. We will need to apply the triangle inequality principle to solve this problem.
Answer
The possible range for the length of side LM is $8 < LM < 26$.
Answer for screen readers
The possible range for the length of side LM is $8 < LM < 26$.
Steps to Solve
- Apply the Triangle Inequality Principle The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have:
- Side LM (unknown length)
- Side MN = 9
- Side LN = 17
This gives us the following inequalities:
$$ LM + 9 > 17 $$ $$ LM + 17 > 9 $$ $$ 9 + 17 > LM $$
- Simplify the inequalities Now we will simplify each inequality:
-
For the first inequality: $$ LM + 9 > 17 $$ Subtracting 9 from both sides gives: $$ LM > 8 $$
-
For the second inequality: $$ LM + 17 > 9 $$ Subtracting 17 from both sides gives: $$ LM > -8 $$ (This inequality doesn’t provide a new constraint since lengths must be positive.)
-
For the third inequality: $$ 9 + 17 > LM $$ Simplifying gives: $$ 26 > LM \text{ or } LM < 26 $$
- Combine the inequalities From the valid inequalities, we have:
$$ LM > 8 \quad \text{and} \quad LM < 26 $$
This can be combined into a single compound inequality:
$$ 8 < LM < 26 $$
The possible range for the length of side LM is $8 < LM < 26$.
More Information
This solution relies on the triangle inequality principle, which is fundamental in geometry. It ensures that the lengths of triangle sides satisfy specific relationships, necessary for forming a valid triangle.
Tips
- Misapplying the triangle inequality or overlooking one of the inequalities can lead to incorrect ranges. It’s essential to include all three inequalities and check that they reinforce the same constraints.
AI-generated content may contain errors. Please verify critical information
