7.1 Q-8

Question Statement

Given the points $B(4, 1)$, $C(-2, 3)$, and $D(-8, 0)$, use the vector method to find the coordinates of the following points:

(i) Point $A$, if $A B C D$ is a parallelogram.
(ii) Point $E$, if $A E B D$ is a parallelogram.

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Background and Explanation

In this problem, we need to use the vector method to find the coordinates of unknown points in a parallelogram. The key properties of a parallelogram we’ll use are:

1. Opposite sides are equal and parallel. For example, if $A B C D$ is a parallelogram, then the vector $A B$ is equal to the vector $D C$ and vice versa.
2. Vector addition: The vector sum of two adjacent sides of a parallelogram should result in the vector across the diagonal.

By applying these principles, we can solve for the coordinates of points $A$ and $E$.

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Solution

(i) Find the coordinates of point $A$, given that $A B C D$ is a parallelogram

1. Set up the equation:
   Since $A B C D$ is a parallelogram, we know that $A B = D C$.
   The vector $A B$ is the difference between the coordinates of $B(4, 1)$ and $A(x, y)$, and the vector $D C$ is the difference between the coordinates of $D(-8, 0)$ and $C(-2, 3)$.
   Thus: $A B = D C$
   This translates to the equation: $(4, 1) - (x, y) = (-2, 3) - (-8, 0)$

2. Simplify the equation:
   Subtract the coordinates:
   $(4 - x, 1 - y) = (6, 3)$

3. Solve for $x$ and $y$:
   By comparing the components:

   • $4 - x = 6$ gives $x = -2$
   • $1 - y = 3$ gives $y = -2$

Thus, the coordinates of point $A$ are $(-2, -2)$.

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(ii) Find the coordinates of point $E$, given that $A E B D$ is a parallelogram

1. Set up the equation:
   Since $A E B D$ is a parallelogram, we know that $A E = D B$.
   The vector $A E$ is the difference between the coordinates of $E(x, y)$ and $A(-2, -2)$, and the vector $D B$ is the difference between the coordinates of $D(-8, 0)$ and $B(4, 1)$.
   Thus: $A E = D B$
   This translates to the equation: $(x, y) - (-2, -2) = (4, 1) - (-8, 0)$

2. Simplify the equation:
   Subtract the coordinates:
   $(x + 2, y + 2) = (12, 1)$

3. Solve for $x$ and $y$:
   By comparing the components:

   • $x + 2 = 12$ gives $x = 10$
   • $y + 2 = 1$ gives $y = -1$

Thus, the coordinates of point $E$ are $(10, -1)$.

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Key Formulas or Methods Used

• Vector Addition/Subtraction:
  The vector between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by:
  $\overrightarrow{P Q} = (x_2 - x_1, y_2 - y_1)$

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

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Summary of Steps

1. For $A B C D$ is a parallelogram:

   • Set up the equation $A B = D C$.
   • Solve for the coordinates of point $A$ using vector subtraction.

2. For $A E B D$ is a parallelogram:

   • Set up the equation $A E = D B$.
   • Solve for the coordinates of point $E$ using vector subtraction.

3. Simplify the equations to find the unknown coordinates.