Question

Let \(A, B,\) and \(C\) denote the three vertice of a triangle. a. If \(E\) is the midpoint of side \(B C\), show that $$ \overrightarrow{A E}=\frac{1}{2}(\overrightarrow{A B}+\overrightarrow{A C}) $$ b. If \(F\) is the midpoint of side \(A C\), show that $$ \overrightarrow{F E}=\frac{1}{2} \overrightarrow{A B} $$

Short answer

a. \(\overrightarrow{AE} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})\); b. \(\overrightarrow{FE} = \frac{1}{2}\overrightarrow{AB}\).

Step-by-step solution

Understanding the Midpoint Formula for Part a

The midpoint of a line segment between two points in vector form is given by the average of the two vectors. Here, point \(E\) is the midpoint of side \(BC\). Thus, \(\overrightarrow{BE} = \overrightarrow{EC}\).

Defining Midpoint E in Vector Terms

To find \(\overrightarrow{AE}\), express \(\overrightarrow{E}\) using vectors \(\overrightarrow{B}\) and \(\overrightarrow{C}\): \(\overrightarrow{E} = \frac{\overrightarrow{B} + \overrightarrow{C}}{2}\). Therefore, \(\overrightarrow{AE} = \overrightarrow{OE} - \overrightarrow{OA} = \frac{1}{2}(\overrightarrow{B} + \overrightarrow{C}) - \overrightarrow{A}\).

Transforming Vector Expressions

Substitute the expressions for \(\overrightarrow{B}\) and \(\overrightarrow{C}\) in terms of \(\overrightarrow{A}\), \(\overrightarrow{B} = \overrightarrow{A} + \overrightarrow{AB}\) and \(\overrightarrow{C} = \overrightarrow{A} + \overrightarrow{AC}\), into the equation: \(\overrightarrow{AE} = \frac{1}{2}((\overrightarrow{A} + \overrightarrow{AB}) + (\overrightarrow{A} + \overrightarrow{AC})) - \overrightarrow{A}\).

Simplifying Vector Expression for AE

Simplify the right-hand side: \(\overrightarrow{AE} = \frac{1}{2}(2\overrightarrow{A} + \overrightarrow{AB} + \overrightarrow{AC}) - \overrightarrow{A} = \overrightarrow{A} + \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC}) - \overrightarrow{A}\). This simplifies to \(\overrightarrow{AE} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC}) \).

Understanding the Midpoint Formula for Part b

Point \(F\) is the midpoint of side \(AC\), hence \(\overrightarrow{AF} = \overrightarrow{FC}\). We need to express \(\overrightarrow{FE}\).

Defining Midpoint F in Vector Terms

Express \(\overrightarrow{F}\) using \(\overrightarrow{A}\) and \(\overrightarrow{C}\): \(\overrightarrow{F} = \frac{\overrightarrow{A} + \overrightarrow{C}}{2}\). Therefore, \(\overrightarrow{FE} = \overrightarrow{E} - \overrightarrow{F}\).

Expressing and Simplifying FE

Substitute the expressions for \(\overrightarrow{E}\) and \(\overrightarrow{F}\): \(\overrightarrow{FE} = \left(\frac{\overrightarrow{B} + \overrightarrow{C}}{2}\right) - \left(\frac{\overrightarrow{A} + \overrightarrow{C}}{2}\right)\). Simplify to get: \(\overrightarrow{FE} = \frac{1}{2}\overrightarrow{B} - \frac{1}{2}\overrightarrow{A}\), which simplifies to \(\overrightarrow{FE} = \frac{1}{2}(\overrightarrow{B} - \overrightarrow{A}) = \frac{1}{2}\overrightarrow{AB}\).

Key concepts

Midpoint Formula

The midpoint formula is a fundamental concept in vector geometry. This formula helps in finding a point that lies exactly in the middle of a line segment. Given two points, say point \(B\) with position vector \(\overrightarrow{B}\) and point \(C\) with position vector \(\overrightarrow{C}\), the midpoint \(E\), lying on the line segment joining these two points, can be calculated using:\[ \overrightarrow{E} = \frac{\overrightarrow{B} + \overrightarrow{C}}{2} \]

• This translates to finding the average of the vectors \(\overrightarrow{B}\) and \(\overrightarrow{C}\).
• The logic is simple: the midpoint is equidistant from both endpoints.

In our exercise, when \(E\) is the midpoint of side \(BC\), using the formula helps simplify expressions like \(\overrightarrow{AE}\). By expressing \(\overrightarrow{E}\) in its vector form, we can subtract the vector of point \(A\) to find \(\overrightarrow{AE}\). This provides a clear and concise method to derive other vector relations.

Vector Addition

Vector addition is a crucial concept that allows us to analyze and solve problems involving multiple vectors. It involves adding two or more vectors to get a resultant vector. When applied, it can help in finding solutions for the positions and the directions of different points in a geometric figure like a triangle.

• To add vectors, simply add their corresponding components. For instance, to add vectors \(\overrightarrow{A} = (x_1, y_1)\) and \(\overrightarrow{B} = (x_2, y_2)\), the sum is \(\overrightarrow{A} + \overrightarrow{B} = (x_1 + x_2, y_1 + y_2)\).
• The result represents a new vector, termed as the resultant vector, pointing from the initial position to the terminal position.

In our exercise, vector addition is used to express \(\overrightarrow{E}\) and \(\overrightarrow{F}\) as the result of averaging two position vectors. This highlights its significance in breaking down complex geometric situations into simple vector equations that can easily be manipulated to find desired relationships.

Triangle Vertices

Vertices of a triangle in vector geometry are often described using position vectors that extend from the origin to a point. The overall shape and size of a triangle can be described by these vectors, and relationships between them can reveal important properties about the triangle itself.

• The vertices, labeled as \(A\), \(B\), and \(C\), make up the triangle. Their position vectors are usually given as \(\overrightarrow{A}\), \(\overrightarrow{B}\), and \(\overrightarrow{C}\) respectively.
• Understanding these base points lets us calculate side vectors like \(\overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A}\) and \(\overrightarrow{AC} = \overrightarrow{C} - \overrightarrow{A}\) with ease, which are crucial for solving complex geometrical problems.

In the context of our problem, understanding the role of these vertices means we can work out midpoint and vector expressions such as \(\overrightarrow{AE}\) or \(\overrightarrow{FE}\). It allows us to explore and prove geometric relationships within the triangle, like the connection between midpoints and vertex-based vectors.