Question

If \( L, M, N \) are the midpoints of the sides PQ, QR, and RP of triangle \( \Delta PQR \), then \( \overline{QM} + \overline{LN} + \overline{ML} + \overline{RN} - \overline{MN} - \overline{QL} = \):

• A. \( \overline{PQ} + \overline{QR} + \overline{LM} + \overline{MN} \)
• B. \( \overline{LP} + \overline{PM} + \overline{MQ} \)
• C. \( \overline{PQ} + \overline{QR} - \overline{PR} \)
• D. \( \overline{LM} + \overline{MN} + \overline{NR} \)

Hint:

For problems involving midpoints and geometric figures, utilize the symmetry of the figure and properties like the midpoint theorem to reduce the problem to simpler terms.

Answer 1

The Correct Option is C

We are given a triangle \( \Delta PQR \) with points \( L, M, N \) as the midpoints of the sides: \[ \overline{L} \text{ (Midpoint of } \overline{PQ}), \quad \overline{M} \text{ (Midpoint of } \overline{QR}), \quad \overline{N} \text{ (Midpoint of } \overline{RP}) \] We need to evaluate the expression: \[ \overline{QM} + \overline{LN} + \overline{ML} + \overline{RN} - \overline{MN} - \overline{QL} \] Step 1: Identify Midpoint Properties
By the midpoint theorem: \[ \overline{LN} = \frac{1}{2} \overline{PR}, \quad \overline{ML} = \frac{1}{2} \overline{PQ}, \quad \overline{MN} = \frac{1}{2} \overline{QR} \] Also, \[ \overline{QM} = \frac{1}{2} \overline{QR}, \quad \overline{RN} = \frac{1}{2} \overline{PR}, \quad \overline{QL} = \frac{1}{2} \overline{PQ} \] Step 2: Add and Subtract Terms
Now combine the given expression: \[ \overline{QM} + \overline{LN} + \overline{ML} + \overline{RN} - \overline{MN} - \overline{QL} \] Substituting the midpoint values: \[ = \frac{1}{2} \overline{QR} + \frac{1}{2} \overline{PR} + \frac{1}{2} \overline{PQ} + \frac{1}{2} \overline{PR} - \frac{1}{2} \overline{QR} - \frac{1}{2} \overline{PQ} \] Step 3: Simplifying
By combining like terms: - \( \frac{1}{2} \overline{QR} - \frac{1}{2} \overline{QR} = 0 \) - \( \frac{1}{2} \overline{PQ} - \frac{1}{2} \overline{PQ} = 0 \) - Remaining terms: \[ = \frac{1}{2} \overline{PR} + \frac{1}{2} \overline{PR} = \overline{PR} \] Now recall the identity in triangle geometry: \[ \overline{PQ} + \overline{QR} - \overline{PR} \] Step 4: Final Answer
\[ \boxed{\overline{PQ} + \overline{QR} - \overline{PR}} \] Final Answer: (C) \( \overline{PQ} + \overline{QR} - \overline{PR} \)