Question

Use vectors to prove the following theorems from geometry; In a parallelogram, the two lines from one corner to the midpoints of the two opposite sides trisect the diagonal they cross,

Short answer

The two lines trisect the diagonal.

Step-by-step solution

Define the Parallelogram

Consider a parallelogram ABCD with vertices A, B, C, and D. Let the midpoints of sides BC and AD be M and N respectively.

Midpoints Representation

Represent the midpoints M and N using vectors. If A is the origin, then B is represented by \(\textbf{b}\), C is \(\textbf{b} + \textbf{d}\), and D is \(\textbf{d}\). The midpoint M of BC is \(M = \frac{\textbf{b} + (\textbf{b} + \textbf{d})}{2} = \textbf{b} + \frac{\textbf{d}}{2}\). Similarly, the midpoint N of AD is \(N = \frac{\textbf{d}}{2}\).

Equation of the Line AN

Find the equation of line AN. The line from A to midpoint N can be parameterized as \( \textbf{r}(t) = t \frac{\textbf{d}}{2}\), where \(0 \leq t \leq 2\).

Equation of the Line CM

Now find the equation of line CM. The line from C to midpoint M can be parameterized as \( \textbf{r'}(t) = \textbf{b} + \textbf{d} + t \frac{-\textbf{b} + \frac{\textbf{d}}{2}}{2} = \textbf{b} + \textbf{d} + t (-\frac{\textbf{b}}{2} + \frac{\textbf{d}}{4})\), where \(0 \leq t \leq 2\).

Intersection Point

Find the intersection of lines AN and CM. Equate the parameterized vectors to find the value of t where they intersect. \(\textbf{b} + \textbf{d} + t(-\frac{\textbf{b}}{2} + \frac{\textbf{d}}{4}) = \frac{\textbf{d}}{2}s\). Solve for \(s\).

Verification

Verify the intersection point lies on the diagonal AC. Find the coordinates of the intersection and confirm it divides AC in the ratio 1:2.

Key concepts

parallelogram

A parallelogram is a four-sided figure with opposite sides parallel and equal in length. Parallelograms have special properties which make them useful in vector geometry. These properties include the fact that opposite angles are equal and the diagonals bisect each other. In this exercise, we use the properties of parallelograms to find relationships between points and lines within the shape.

• All sides in a parallelogram have parallel opposite sides.
• Opposite angles are equal.
• The diagonals bisect each other.

Understanding these properties help us navigate the geometric problem effectively.

trisection

Trisection means dividing something into three equal parts. In vector geometry, we often use the concept of trisection to find specific points along a line or shape. For example, to trisect a diagonal of a parallelogram using vectors, we need to define specific points along the diagonal which divide the line into three equal segments.

• The point of trisection are often found using ratios.
• Vectors can represent these points where the lines intersect the diagonal.

For instance, identifying where a line from a vertex meets a midpoint, and work out how it splits another line into three parts.

midpoint

Midpoints are crucial in this exercise as they form part of the lines we analyze. A midpoint of a line segment between two points is simply the average of their coordinates. Using vector notation, if point A is the origin, then midpoint M of line segment BC can be found by averaging the vectors representing B and C.

• Midpoint of two points (A and B) is \(M = \frac{A + B}{2}\).
• Vectors can streamline calculating midpoints, which is essential in our parallelogram problem.

This provides a position we can use to form other vectors and ultimately solve the exercise.

vector equations

Vector equations are tools to describe lines in space. They employ a parameter to trace points along a line. For example, the line AN can be represented as \( \textbf{r}(t) = t \frac{\textbf{d}}{2} \), where t is the parameter ranging from 0 to 2. Similarly, the line CM is parameterized.

• A vector equation for a line has the form \( \textbf{r} = \textbf{A} + t\textbf{B} \), where A is a point on the line and B is a direction vector.
• Parametric form helps us find specific points along the line by varying t.

For solving intersection problems, equate the parameterized vectors to find common points.

intersection point

Finding the intersection point of two lines means finding a common point shared by both lines. In vector geometry, we use the parametric equations of the lines and solve to find the vector parameter at which they intersect. For example, to find where lines AN and CM of the parallelogram intersect, equate their vector equations.

• Substitute the parameterized forms and solve for the parameter to find the intersection point.
• Plug the parameter value back into the vector equation to get precise coordinates.

This verifies the point lies on both lines and helps confirm other geometric properties, like trisecting the diagonal.