4.3 Q-27

Question Statement

Find the three remaining vertices of a parallelogram if one vertex is $A(1, 4)$, the diagonals intersect at $(2, 1)$, and the sides have slopes of $1$ and $-1$.

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Background and Explanation

A parallelogram has the following properties:

1. Diagonals bisect each other: The point where the diagonals intersect is the midpoint of both diagonals.
2. Opposite sides are parallel and have the same slope: This is useful for finding equations of lines connecting the vertices.
3. Coordinate Geometry Basics: The slope formula and midpoint formula will be applied here:
   • Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
   • Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

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Solution

Let the vertices of the parallelogram be $A(1, 4)$, $B(a, b)$, $C(c, d)$, and $D(e, f)$. The diagonals intersect at $M(2, 1)$, which is the midpoint of both diagonals.

Step 1: Using the Midpoint Formula

Using $M$ as the midpoint of $\overline{AC}$, we have:

$$\frac{1 + c}{2} = 2 \quad \text{and} \quad \frac{4 + d}{2} = 1$$

Solving for $c$ and $d$:

$$1 + c = 4 \quad \Rightarrow \quad c = 3$$ $$4 + d = 2 \quad \Rightarrow \quad d = -2$$

Thus, vertex $C$ is $(3, -2)$.

Step 2: Using the Slope of the Sides

The sides of the parallelogram have slopes $1$ and $-1$. Using this information:

• Slope of $\overline{AD} = 1$:

$$\frac{f - 4}{e - 1} = 1 \quad \Rightarrow \quad f - 4 = e - 1 \quad \Rightarrow \quad e - f + 3 = 0 \tag{1}$$

• Slope of $\overline{BC} = 1$:

$$\frac{d - b}{c - a} = 1 \quad \Rightarrow \quad \frac{-2 - b}{3 - a} = 1 \quad \Rightarrow \quad -2 - b = a - 3 \quad \Rightarrow \quad a - b - 5 = 0 \tag{2}$$

• Slope of $\overline{AB} = -1$:

$$\frac{b - 4}{a - 1} = -1 \quad \Rightarrow \quad b - 4 = -a + 1 \quad \Rightarrow \quad a + b - 5 = 0 \tag{3}$$

• Slope of $\overline{CD} = -1$:

$$\frac{f - d}{e - c} = -1 \quad \Rightarrow \quad \frac{f + 2}{e - 3} = -1 \quad \Rightarrow \quad f + 2 = -e + 3 \quad \Rightarrow \quad e + f + 11 = 0 \tag{4}$$

Step 3: Solving the Equations

We now solve the system of equations:

1. From (2) and (3):

$$a - b - 5 = 0 \quad \text{and} \quad a + b - 5 = 0$$

Adding the equations:

$$2a = 10 \quad \Rightarrow \quad a = 5$$

Substituting $a = 5$ into (2):

$$5 - b - 5 = 0 \quad \Rightarrow \quad b = 3$$

Thus, vertex $B$ is $(5, 3)$.

2. From (1) and (4):

$$e - f + 3 = 0 \quad \text{and} \quad e + f + 11 = 0$$

Adding the equations:

$$2e + 14 = 0 \quad \Rightarrow \quad e = -7$$

Substituting $e = -7$ into (1):

$$-7 - f + 3 = 0 \quad \Rightarrow \quad f = -4$$

Thus, vertex $D$ is $(-7, -4)$.

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Key Formulas or Methods Used

• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Properties of a Parallelogram:
  • Diagonals bisect each other.
  • Opposite sides are parallel and have the same slope.

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Summary of Steps

1. Use the midpoint formula to find $C(3, -2)$.
2. Use the slope condition to establish equations for $A, B, C, D$.
3. Solve the system of equations to find:
   • $B(5, 3)$
   • $D(-7, -4)$.