Find The Perimeter Of The Parallelogram Whose Vertices Are A (-1,1), B (-9,6), C(-12,11) And D(-4,6)

Find The Perimeter Of The Parallelogram Whose Vertices Are A (-1,1), B (-9,6), C(-12,11) And D(-4,6)

Understanding how to find the perimeter of a parallelogram when given the coordinates of its vertices is a fundamental skill in coordinate geometry. In this article, we will go through a detailed, step-by-step process to determine the perimeter of the parallelogram with vertices A (-1, 1), B (-9, 6), C (-12, 11), and D (-4, 6). We will explore concepts such as distance formula, properties of parallelograms, and coordinate calculations, ensuring that you not only find the answer but also understand the underlying principles involved.

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Understanding the Problem: Coordinates and Parallelogram Properties

Before diving into calculations, it’s essential to understand what is being asked and the properties of the shapes involved.

Vertices of the Parallelogram

• Vertex A: (-1, 1)
• Vertex B: (-9, 6)
• Vertex C: (-12, 11)
• Vertex D: (-4, 6)

These points are given in a coordinate plane, and the goal is to find the total perimeter of the parallelogram formed by these points.

Properties of a Parallelogram

• Opposite sides are equal in length.
• Opposite sides are parallel.
• The diagonals bisect each other.

Knowing these properties helps verify the shape and guide which pairs of points form sides.

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Verifying the Shape: Are the Given Points a Parallelogram?

Before calculating the perimeter, we should confirm that the points indeed form a parallelogram.

Method 1: Check if Opposite Sides are Parallel

• Calculate the slopes of the sides to see if opposite sides are parallel.
• Use the slope formula: \( m = \frac{y2 - y1}{x2 - x1} \).

Calculating Slopes of Sides

• AB: between points A (-1, 1) and B (-9, 6)

\[ m_{AB} = \frac{6 - 1}{-9 - (-1)} = \frac{5}{-8} = -\frac{5}{8} \]

• BC: between B (-9, 6) and C (-12, 11)

\[ m_{BC} = \frac{11 - 6}{-12 - (-9)} = \frac{5}{-3} = -\frac{5}{3} \]

• CD: between C (-12, 11) and D (-4, 6)

\[ m_{CD} = \frac{6 - 11}{-4 - (-12)} = \frac{-5}{8} = -\frac{5}{8} \]

• DA: between D (-4, 6) and A (-1, 1)

\[ m_{DA} = \frac{1 - 6}{-1 - (-4)} = \frac{-5}{3} = -\frac{5}{3} \]

Analysis of Slopes

• Sides AB and CD have slopes of \(-\frac{5}{8}\), indicating they are parallel.
• Sides BC and DA have slopes of \(-\frac{5}{3}\), indicating they are parallel.

Since opposite sides are parallel, the shape is indeed a parallelogram.

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Calculating the Lengths of Sides

To find the perimeter, we need the lengths of two adjacent sides. We will use the distance formula:

\[
d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\]

Calculating Side Lengths

1. Length of AB

\[ AB = \sqrt{(-9 - (-1))^2 + (6 - 1)^2} = \sqrt{(-8)^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89} \]

1. Length of BC

\[ BC = \sqrt{(-12 - (-9))^2 + (11 - 6)^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]

1. Length of CD

\[ CD = \sqrt{(-4 - (-12))^2 + (6 - 11)^2} = \sqrt{8^2 + (-5)^2} = \sqrt{64 + 25} = \sqrt{89} \]

1. Length of DA

\[ DA = \sqrt{(-1 - (-4))^2 + (1 - 6)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

Observation:

• AB and CD are equal in length (\(\sqrt{89}\))


• BC and DA are equal in length (\(\sqrt{34}\))

This confirms the shape's property of equal opposite sides, characteristic of a parallelogram.

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Calculating the Perimeter

The perimeter \(P\) of a parallelogram is given by:

\[
P = 2 \times (\text{length of one side}) + 2 \times (\text{length of adjacent side})
\]

Using the side lengths calculated:

\[
P = 2 \times AB + 2 \times BC = 2 \times \sqrt{89} + 2 \times \sqrt{34}
\]

Simplify:

\[
P = 2 (\sqrt{89} + \sqrt{34})
\]

For a numerical approximation:

\[
\sqrt{89} \approx 9.43
\]
\[
\sqrt{34} \approx 5.83
\]

Therefore,

\[
P \approx 2 (9.43 + 5.83) = 2 \times 15.26 = 30.52
\]

Final answer:
\[
\boxed{
\text{Perimeter} \approx 30.52 \text{ units}
}
\]

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Summary and Key Takeaways

• The vertices provided form a parallelogram, verified by the slopes of opposite sides.
• Calculated side lengths using the distance formula: \(\sqrt{89}\) and \(\sqrt{34}\).
• The perimeter is twice the sum of the lengths of adjacent sides.
• The approximate perimeter of the parallelogram is 30.52 units.

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Additional Tips for Coordinate Geometry Problems

• Always verify the shape using properties such as parallel sides or diagonals bisecting each other.
• Use the distance formula carefully, and double-check calculations.
• When exact values are preferred, leave the answer in terms of square roots; for approximate numerical answers, use calculator approximations.
• Visualizing the shape on a graph can help better understand the problem.

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Conclusion

Finding the perimeter of a parallelogram from its vertices involves understanding properties of the shape, verifying the shape with slopes, calculating side lengths with the distance formula, and then applying the perimeter formula. For the given vertices A (-1, 1), B (-9, 6), C (-12, 11), and D (-4, 6), the perimeter is approximately 30.52 units. Mastering these steps enhances your ability to solve similar geometry problems efficiently and accurately.