7.1 Q-14

Question Statement

Prove that the line segments joining the midpoints of the sides of a quadrilateral, taken in order, form a parallelogram.

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Background and Explanation

To prove that the quadrilateral formed by the midpoints of the sides of another quadrilateral is a parallelogram, we need to demonstrate that the opposite sides of the quadrilateral formed by these midpoints are parallel and equal in length.

The key concept here is the midpoint theorem, which states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. We will apply this idea to a quadrilateral.

We also need to use the vector method to express the position vectors of the vertices of the quadrilateral and the midpoints, then show that the opposite sides of the quadrilateral formed by these midpoints are both parallel and equal in length.

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Solution

Step 1: Define the vertices of the quadrilateral

Let the vertices of the quadrilateral be $A, B, C, D$, with their respective position vectors denoted as:

• $\mathbf{a}$ for $A$,
• $\mathbf{b}$ for $B$,
• $\mathbf{c}$ for $C$,
• $\mathbf{d}$ for $D$.

Step 2: Define the midpoints of the sides

Let the midpoints of the sides $A B$, $B C$, $C D$, and $D A$ be $E, F, G,$ and $H$ respectively. The position vectors of these midpoints are given by:

• Position vector of $E$ (midpoint of $A B$):
  $\mathbf{E} = \frac{\mathbf{a} + \mathbf{b}}{2}$

• Position vector of $F$ (midpoint of $B C$):
  $\mathbf{F} = \frac{\mathbf{b} + \mathbf{c}}{2}$

• Position vector of $G$ (midpoint of $C D$):
  $\mathbf{G} = \frac{\mathbf{c} + \mathbf{d}}{2}$

• Position vector of $H$ (midpoint of $D A$):
  $\mathbf{H} = \frac{\mathbf{d} + \mathbf{a}}{2}$

Step 3: Show that $E F \parallel H G$

To show that $E F \parallel H G$, we calculate the vectors $E \vec{F}$ and $H \vec{G}$.

1. Find the vector $E \vec{F}$: $E \vec{F} = \mathbf{F} - \mathbf{E} = \frac{\mathbf{b} + \mathbf{c}}{2} - \frac{\mathbf{a} + \mathbf{b}}{2} = \frac{\mathbf{c} - \mathbf{a}}{2}$

2. Find the vector $H \vec{G}$: $H \vec{G} = \mathbf{G} - \mathbf{H} = \frac{\mathbf{c} + \mathbf{d}}{2} - \frac{\mathbf{d} + \mathbf{a}}{2} = \frac{\mathbf{c} - \mathbf{a}}{2}$

Since $E \vec{F} = H \vec{G}$, we conclude that $E F \parallel H G$ and both are equal in length.

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Step 4: Show that $E H \parallel F G$

Next, we calculate the vectors $H \vec{E}$ and $G \vec{F}$ to show that $E H \parallel F G$.

1. Find the vector $H \vec{E}$: $H \vec{E} = \mathbf{E} - \mathbf{H} = \frac{\mathbf{a} + \mathbf{b}}{2} - \frac{\mathbf{d} + \mathbf{a}}{2} = \frac{\mathbf{b} - \mathbf{d}}{2}$

2. Find the vector $G \vec{F}$: $G \vec{F} = \mathbf{F} - \mathbf{G} = \frac{\mathbf{b} + \mathbf{c}}{2} - \frac{\mathbf{c} + \mathbf{d}}{2} = \frac{\mathbf{b} - \mathbf{d}}{2}$

Since $H \vec{E} = G \vec{F}$, we conclude that $E H \parallel F G$ and both are equal in length.

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Step 5: Conclude that $E F G H$ is a parallelogram

Since both pairs of opposite sides $E F \parallel H G$ and $E H \parallel F G$ are parallel and equal in length, we can conclude that the quadrilateral $E F G H$ formed by the midpoints of the sides of quadrilateral $A B C D$ is a parallelogram.

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Key Formulas or Methods Used

• Midpoint Formula:
  The position vector of the midpoint of a line segment joining two points $A$ and $B$ is:
  $\text{Midpoint} = \frac{\mathbf{a} + \mathbf{b}}{2}$

• Vector Subtraction:
  The vector from point $A(x_1, y_1)$ to point $B(x_2, y_2)$ is:
  $\overrightarrow{A B} = (x_2 - x_1, y_2 - y_1)$

• Parallel Vectors:
  Two vectors are parallel if one is a scalar multiple of the other.

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Summary of Steps

1. Find the position vectors of the midpoints of the sides of the quadrilateral.
2. Use vector subtraction to find the vectors joining the midpoints.
3. Show that the opposite sides $E F \parallel H G$ and $E H \parallel F G$ are both parallel and equal in length.
4. Conclude that the quadrilateral formed by the midpoints is a parallelogram.