Problem related to section formula

Proving the Midpoint Line Theorem in a Triangle Using Vector Method

Problem Statement: Prove using vectors that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and has half its length.

Given:

Let O be the origin of reference in a coordinate system. Consider a triangle ABC where the position vectors of the vertices are:
\( \overrightarrow{OA} = \overrightarrow{a} \), \( \overrightarrow{OB} = \overrightarrow{b} \), \( \overrightarrow{OC} = \overrightarrow{c} \)

Let point D be the midpoint of side AB:
\( \overrightarrow{OD} = \frac{\overrightarrow{a} + \overrightarrow{b}}{2} \)

Let point E be the midpoint of side AC:
\( \overrightarrow{OE} = \frac{\overrightarrow{a} + \overrightarrow{c}}{2} \)

To Prove:

• \( \overrightarrow{DE} \parallel \overrightarrow{BC} \)
• \( \overrightarrow{DE} = \frac{1}{2} \overrightarrow{BC} \)

Proof Using Vectors:

Find the vector from D to E:
\( \overrightarrow{DE} = \overrightarrow{OE} - \overrightarrow{OD} \)

Substitute the values:
\( \overrightarrow{DE} = \left( \frac{\overrightarrow{a} + \overrightarrow{c}}{2} \right) - \left( \frac{\overrightarrow{a} + \overrightarrow{b}}{2} \right) \)

Simplify:
\( \overrightarrow{DE} = \frac{\overrightarrow{c} - \overrightarrow{b}}{2} \)

Now compute vector \( \overrightarrow{BC} \):
\( \overrightarrow{BC} = \overrightarrow{OC} - \overrightarrow{OB} = \overrightarrow{c} - \overrightarrow{b} \)

Hence:
\( \overrightarrow{DE} = \frac{1}{2} \overrightarrow{BC} \)

Since \( \overrightarrow{DE} \) is a scalar multiple of \( \overrightarrow{BC} \), the vectors are parallel. The scalar factor \( \frac{1}{2} \) also confirms that the length of \( \overrightarrow{DE} \) is half the length of \( \overrightarrow{BC} \).

Therefore, it is proved: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length.