7.1 Q-7

Question Statement

Given the points $A(2, -4)$, $B(4, 0)$, and $C(1, 6)$, use the vector method to find the coordinates of point $D$ if:

i. $A B C D$ is a parallelogram
ii. $A D B C$ is a parallelogram

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Background and Explanation

In this problem, we are asked to find the coordinates of point $D$ given certain conditions about the geometry of the points. We use the properties of a parallelogram and the vector method to solve the problem.

• Parallelogram Property: In a parallelogram, opposite sides are equal and parallel. This means for the parallelogram $A B C D$, the vector $\overrightarrow{A B}$ is equal to the vector $\overrightarrow{C D}$, and similarly for the other pair of opposite sides.

• Vector Addition: To find the unknown point $D$, we use the fact that the sum of the vectors along the sides of the parallelogram should lead us back to the same point.

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Solution

(i) $A B C D$ is a parallelogram

1. Set up the relationship between the vectors:
   Since $A B C D$ is a parallelogram, the vector $A B$ is equal to the vector $D C$. This gives us the equation: $A B = D C$
   So,
   $(4, 0) - (2, -4) = (1, 6) - (x, y)$

2. Simplify the equation:
   Subtract the coordinates:
   $(4 - 2, 0 + 4) = (1 - x, 6 - y)$
   This simplifies to:
   $(2, 4) = (1 - x, 6 - y)$

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

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

Thus, the coordinates of point $D$ are $(-1, 2)$.

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(ii) $A D B C$ is a parallelogram

1. Set up the relationship between the vectors:
   Since $A D B C$ is a parallelogram, the vector $A D$ is equal to the vector $C B$. Additionally, the vector $\overrightarrow{A D}$ is parallel to $\overrightarrow{C D}$.
   $A D = C B$
   This gives us the equation:
   $(x, y) - (2, -4) = (4, 0) - (1, 6)$

2. Simplify the equation:
   Subtract the coordinates:
   $(x - 2, y + 4) = (4 - 1, 0 - 6)$
   This simplifies to:
   $(x - 2, y + 4) = (3, -6)$

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

   • $x - 2 = 3$ gives $x = 5$
   • $y + 4 = -6$ gives $y = -10$

Thus, the coordinates of point $D$ are $(5, -10)$.

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Key Formulas or Methods Used

• Vector Addition/Subtraction:
  The vector from one point to another is given by the difference in their coordinates: $\overrightarrow{A B} = (x_B - x_A, y_B - y_A)$

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

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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 $D$ using vector subtraction.

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

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

3. Simplify and solve the resulting system of equations for $x$ and $y$.