To find the perimeter of the parallelogram formed by the given vertices (2,3), (4,4), and (4,2), we first need to determine all four vertices and the lengths of its sides.
The given vertices can be used to identify the other vertex. Let's label the points:
- \( A(2, 3) \)
- \( B(4, 4) \)
- \( C(4, 2) \)
We can find the fourth vertex \( D \) by using the property of a parallelogram that the midpoints of the diagonals are the same.
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Calculate the midpoint of diagonal AC: \[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{2 + 4}{2}, \frac{3 + 2}{2} \right) = \left( 3, 2.5 \right) \]
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Calculate the midpoint of diagonal BD: \[ M_{BD} = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) \]
Since \( M_{AC} = M_{BD} \), we have: \[ \left( 3, 2.5 \right) = \left( \frac{4 + x_D}{2}, \frac{4 + y_D}{2} \right) \]
This gives us two equations: \[ \frac{4 + x_D}{2} = 3 \quad \text{ and } \quad \frac{4 + y_D}{2} = 2.5 \]
Solving these equations:
- For \( x_D \): \[ 4 + x_D = 6 \quad \Rightarrow \quad x_D = 2 \]
- For \( y_D \): \[ 4 + y_D = 5 \quad \Rightarrow \quad y_D = 1 \]
Thus the fourth vertex \( D \) is \( (2, 1) \).
Now we have all four vertices of the parallelogram:
- \( A(2, 3) \)
- \( B(4, 4) \)
- \( C(4, 2) \)
- \( D(2, 1) \)
Next, we find the lengths of the sides \( AB \), \( BC \), \( CD \), and \( DA \).
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Length of \( AB \): \[ AB = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
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Length of \( BC \): \[ BC = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + (-2)^2} = \sqrt{4} = 2 \]
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Length of \( CD \): \[ CD = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
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Length of \( DA \): \[ DA = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + (2)^2} = \sqrt{4} = 2 \]
The perimeter \( P \) of the parallelogram is given by: \[ P = AB + BC + CD + DA = \sqrt{5} + 2 + \sqrt{5} + 2 = 2\sqrt{5} + 4 \]
Calculating \( P \) numerically: \[ P \approx 2(2.236) + 4 = 4.472 + 4 = 8.472 \]
Rounding to the nearest tenth: \[ P \approx 8.5 \text{ units} \]
Thus, the perimeter is approximately 8.5 units. Therefore, the answer is: c. 8.5