Question

If D and E are the midpoints of the sides BA and BC of triangle ABC, then \(\overline{AE}+\overline{DC} =\)

• A. \(\overline{AC}\)
• B. \(\frac{3}{2}\overline{BC}\)
• C. \(\frac{3}{2}\overline{AC}\)
• D. \(\frac{1}{2}\overline{AC}\)

Hint:

Use position vectors with midpoints, then add AE and DC. The b terms cancel and 3/2 (c - a) remains.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
In triangle ABC, D is the midpoint of BA and E is the midpoint of BC. We need to express \(\overline{AE} + \overline{DC}\) in terms of the sides.

Step 2: Key Formula or Approach:
Use position vectors \(\vec a, \vec b, \vec c\) for A, B, C. A midpoint has the position vector half the sum of its endpoints. The vector from P to Q is \(\vec q - \vec p\).

Step 3: Midpoints:
\[ \vec e = \frac{\vec b + \vec c}{2}, \qquad \vec d = \frac{\vec a + \vec b}{2} \]

Step 4: Find each vector:
\[ \overline{AE} = \vec e - \vec a = \frac{\vec b + \vec c - 2\vec a}{2} \] \[ \overline{DC} = \vec c - \vec d = \frac{2\vec c - \vec a - \vec b}{2} \]

Step 5: Add:
\[ \overline{AE} + \overline{DC} = \frac{3\vec c - 3\vec a}{2} = \frac32(\vec c - \vec a) = \frac32\overline{AC} \] The \(\vec b\) terms cancel.

Step 6: Check the others:
Option (A) \(\overline{AC}\) would need a factor 1, but we found 3/2. Option (B) is in terms of \(\overline{BC}\), but no \(\vec b\) remains in the sum. Option (D) \(\frac12\overline{AC}\) is also not the factor found.

Final Answer:
The sum equals \(\frac32\overline{AC}\), option (C). \[ \boxed{\frac{3}{2}\overline{AC}} \]