In a triangle, the midsegment is parallel to the third side and is half its length. Given that $RS$ is the midsegment in $\triangle ABC$, we use the midsegment theorem:

$RS = \frac{1}{2} BC$

Given:

• $BC = x + 2$
• $RS = 2x - 14$

Step 1: Set up the equation

$2x - 14 = \frac{1}{2} (x + 2)$

Step 2: Solve for $x$

Multiply both sides by 2 to eliminate the fraction:

$4x - 28 = x + 2$

Subtract $x$ from both sides:

$3x - 28 = 2$

Add 28 to both sides:

$3x = 30$

Divide by 3:

$x = 10$

Step 3: Find the length of $RS$

Substituting $x = 10$ into $RS = 2x - 14$:

$RS = 2(10) - 14 = 20 - 14 = 6$

Final Answer:

• $x = 10$
• $RS = 6$

Would you like a step-by-step explanation with diagrams?

Related Questions:

1. How does the midsegment theorem apply to all triangles?
2. What is the relationship between midsegments and similar triangles?
3. How can you prove the midsegment theorem using coordinate geometry?
4. What happens if a midsegment is given and we need to find the third side?
5. How do midsegments help in triangle area calculations?

Tip:

Always check that your solution satisfies the midsegment theorem by verifying the proportionality between the midsegment and the third side.