7.1 Q-10

Question Statement

Use vectors to prove that quadrilateral $A B C D$ is a parallelogram when the points $A(0,0)$, $B(a, 0)$, $C(b, c)$, and $D(b-a, c)$ are given.

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Background and Explanation

To prove that a quadrilateral is a parallelogram, we need to show that both pairs of opposite sides are equal and parallel. Specifically, for quadrilateral $A B C D$ to be a parallelogram, we need to demonstrate that:

1. $A \vec{B} = D \vec{C}$ (Opposite sides are equal and parallel)
2. $A \vec{D} = B \vec{C}$ (The other pair of opposite sides are equal and parallel)

We use vector subtraction to find the vectors representing the sides of the quadrilateral, then compare them to check if they satisfy the conditions for a parallelogram.

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Solution

Step 1: Find the vector $A \vec{B}$

1. Coordinates of $A$ and $B$:

   • $A(0,0)$
   • $B(a, 0)$

2. Vector $A \vec{B}$:
   $A \vec{B} = (a, 0) - (0, 0) = (a, 0)$
   So, $A \vec{B} = (a, 0)$ (Equation 1).

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Step 2: Find the vector $D \vec{C}$

1. Coordinates of $D$ and $C$:

   • $D(b-a, c)$
   • $C(b, c)$

2. Vector $D \vec{C}$:
   $D \vec{C} = (b, c) - (b-a, c)$
   Simplifying the subtraction: $D \vec{C} = (b - (b-a), c - c) = (a, 0)$
   So, $D \vec{C} = (a, 0)$. This shows that $A \vec{B} = D \vec{C}$, confirming that opposite sides $A B$ and $D C$ are equal and parallel.

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Step 3: Find the vector $A \vec{D}$

1. Coordinates of $A$ and $D$:

   • $A(0, 0)$
   • $D(b-a, c)$

2. Vector $A \vec{D}$:
   $A \vec{D} = (b-a, c) - (0, 0) = (b-a, c)$
   So, $A \vec{D} = (b-a, c)$ (Equation 3).

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Step 4: Find the vector $B \vec{C}$

1. Coordinates of $B$ and $C$:

   • $B(a, 0)$
   • $C(b, c)$

2. Vector $B \vec{C}$:
   $B \vec{C} = (b, c) - (a, 0) = (b-a, c)$
   So, $B \vec{C} = (b-a, c)$ (Equation 4).

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Step 5: Compare the vectors $A \vec{D}$ and $B \vec{C}$

From Equations 3 and 4, we see that:
$A \vec{D} = B \vec{C} = (b-a, c)$

This confirms that the opposite sides $A D$ and $B C$ are equal and parallel.

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Step 6: Conclude that $A B C D$ is a parallelogram

Since both pairs of opposite sides $A B \parallel D C$ and $A D \parallel B C$, the quadrilateral $A B C D$ satisfies the properties of a parallelogram.

Thus, $A B C D$ is a parallelogram.

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Key Formulas or Methods Used

• Vector between two points:
  The vector from point $P(x_1, y_1)$ to point $Q(x_2, y_2)$ is:
  $\overrightarrow{P Q} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}$

• Parallelogram Property:
  Opposite sides of a parallelogram are equal and parallel. Hence:
  $A \vec{B} = D \vec{C} \quad \text{and} \quad A \vec{D} = B \vec{C}$

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Summary of Steps

1. Find the vector $A \vec{B}$ using the coordinates of points $A$ and $B$.
2. Find the vector $D \vec{C}$ using the coordinates of points $D$ and $C$.
3. Show that $A \vec{B} = D \vec{C}$, confirming one pair of opposite sides are equal and parallel.
4. Find the vectors $A \vec{D}$ and $B \vec{C}$.
5. Show that $A \vec{D} = B \vec{C}$, confirming the other pair of opposite sides are equal and parallel.
6. Conclude that $A B C D$ is a parallelogram based on the properties of parallel and equal sides.