If A(2, 3), B(2, 4), C(4, 9), and D(x, 8) are the vertices of a parallelogram ABCD, find the value of
x

Question

If A(2, 3), B(2, 4), C(4, 9), and D(x, 8) are the vertices of a parallelogram ABCD, find the value of 
x

Accepted Answer

#### Step 1: Parallelogram Property
In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal AC must be equal to the midpoint of diagonal BD.

#### Step 2: Midpoint Formula
The midpoint of a line segment with endpoints $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is:  
$$
	ext{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}ight).
$$

#### Step 3: Calculate Midpoint of AC
Given $$ A(2, 3) $$ and $$ C(4, 9) $$:  
$$
	ext{Midpoint of AC} = \left(\frac{2 + 4}{2}, \frac{3 + 9}{2}ight) = (3, 6).
$$

#### Step 4: Calculate Midpoint of BD
Let $$ D(x, 8) $$. Given $$ B(2, 4) $$:  
$$
	ext{Midpoint of BD} = \left(\frac{2 + x}{2}, \frac{4 + 8}{2}ight) = \left(\frac{2 + x}{2}, 6ight).
$$

#### Step 5: Equate Midpoints
Equating the x-coordinates of the midpoints:  
$$
3 = \frac{2 + x}{2}.
$$

Solve for $$ x $$:  
$$
6 = 2 + x.
$$  
$$
x = 4.
$$

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Thus, the value of $$ x $$ is **4**, making Option **(d)** correct.

Answer

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