4.3 Q-6

Question Statement

Given the points $A(7,1)$, $B(-2,2)$, and $C(1,4)$ as consecutive vertices of a parallelogram, find the coordinates of the fourth vertex $D$.

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Background and Explanation

In a parallelogram, opposite sides are both parallel and equal in length. To find the fourth vertex, we can use the property of parallel lines: the slopes of opposite sides are equal. This means that the slope of line $\overline{AB}$ will be equal to the slope of line $\overline{CD}$, and the slope of line $\overline{AD}$ will be equal to the slope of line $\overline{BC}$.

We will use the slope formula to express these relationships and solve for the unknown coordinates of vertex $D$.

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Solution

Step 1: Set up slope equations for parallel sides

We are given the points $A(7,1)$, $B(-2,2)$, and $C(1,4)$. Let the coordinates of the fourth vertex $D$ be $(a, b)$.

1. The slope of $\overline{AB}$ is:

$$m_1 = \frac{2 - 1}{-2 - 7} = \frac{1}{-9} = \frac{-1}{3}$$

2. The slope of $\overline{BC}$ is:

$$m_2 = \frac{4 - 2}{1 - (-2)} = \frac{2}{3}$$

3. The slope of $\overline{AD}$ is:

$$m_3 = \frac{b - 1}{a - 7}$$

4. The slope of $\overline{CD}$ is:

$$m_4 = \frac{4 - b}{1 - a}$$

Step 2: Apply the condition for parallel sides

Since $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \parallel \overline{BC}$, we can set the slopes equal to each other:

1. From $\overline{AB} \parallel \overline{CD}$:

$$m_1 = m_4 \quad \Rightarrow \quad \frac{-1}{3} = \frac{4 - b}{1 - a}$$

Cross-multiply to get the equation:

$$3(4 - b) = -1(1 - a)$$

Simplifying:

$$12 - 3b = -1 + a \quad \Rightarrow \quad a + 3b = 13 \tag{1}$$

2. From $\overline{AD} \parallel \overline{BC}$:

$$m_3 = m_2 \quad \Rightarrow \quad \frac{b - 1}{a - 7} = \frac{2}{3}$$

Cross-multiply to get the equation:

$$3(b - 1) = 2(a - 7)$$

Simplifying:

$$3b - 3 = 2a - 14 \quad \Rightarrow \quad 2a - 3b = 11 \tag{2}$$

Step 3: Solve the system of equations

We now have the system of equations:

1. $a + 3b = 13$
2. $2a - 3b = 11$

To solve for $a$ and $b$, multiply the first equation by 2 and the second equation by 1:

$$2a + 6b = 26 \quad \text{(from equation 1 multiplied by 2)}$$ $$2a - 3b = 11 \quad \text{(from equation 2)}$$

Now subtract the second equation from the first:

$$(2a + 6b) - (2a - 3b) = 26 - 11$$

Simplifying:

$$9b = 15 \quad \Rightarrow \quad b = \frac{15}{9} = 1$$

Step 4: Substitute to find $a$

Substitute $b = 1$ into equation (1):

$$a + 3(1) = 13 \quad \Rightarrow \quad a + 3 = 13 \quad \Rightarrow \quad a = 10$$

Thus, the coordinates of point $D$ are $(a, b) = (10, 1)$.

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Key Formulas or Methods Used

• Slope of a line between two points $(x_1, y_1)$ and $(x_2, y_2)$:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• The slopes of opposite sides of a parallelogram are equal.

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Summary of Steps

1. Calculate the slopes of the sides $\overline{AB}$ and $\overline{BC}$.
2. Set up equations for the slopes of lines $\overline{AD}$ and $\overline{CD}$, ensuring the opposite sides are parallel.
3. Solve the system of equations for the unknown coordinates of point $D$.
4. Find the coordinates of the fourth vertex $D$ as $(10, 1)$.