Course Geometry: Quadrilaterals And Polygons Section Quiz

Course Geometry Unit Quadrilaterals And Polygons Section Squar

A parallelogram has the vertices (0, 3), (3, 0), (0, -3) and (-3, 0). Determine what type of parallelogram and find the perimeter and area. A parallelogram has the vertices (-1, 2), (4, 4), (2, -1) and (-3, -3). Determine what type of parallelogram and find the perimeter and area.

Paper For Above instruction

Introduction

Understanding the properties and classification of parallelograms is fundamental in geometry, especially in the study of quadrilaterals and polygons. This paper analyzes two specific sets of vertices to determine the type of parallelogram they form and computes both their perimeters and areas. These tasks involve applying coordinate geometry principles, including the distance formula for sides, the slope to check for parallelism, and the shoelace theorem for area calculations.

Analysis of the First Parallelogram

The first set of vertices is (0, 3), (3, 0), (0, -3), and (-3, 0). To classify this parallelogram and compute its dimensions, the following steps are undertaken:

1. Calculating Side Lengths:

Using the distance formula—\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)—we compute the lengths of all sides:

- Side between (0, 3) and (3, 0):

\[ d = \sqrt{(3 - 0)^2 + (0 - 3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

- Side between (3, 0) and (0, -3):

\[ d = \sqrt{(0 - 3)^2 + (-3 - 0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

- Side between (0, -3) and (-3, 0):

\[ d = \sqrt{(-3 - 0)^2 + (0 + 3)^2} = \sqrt{9 + 9} = 3\sqrt{2} \]

- Side between (-3, 0) and (0, 3):

\[ d = \sqrt{(0 + 3)^2 + (3 - 0)^2} = \sqrt{9 + 9} = 3\sqrt{2} \]

All sides have equal lengths, indicating that the figure is a rhombus, which is a specific type of parallelogram with all sides equal.

2. Checking for Right Angles:

To determine if it is a square, check for right angles via the slopes of adjacent sides; slopes should be negative reciprocals:

- Slope between (0, 3) and (3, 0):

\[ m_1 = \frac{0 - 3}{3 - 0} = -1 \]

- Slope between (3, 0) and (0, -3):

\[ m_2 = \frac{-3 - 0}{0 - 3} = \frac{-3}{-3} = 1 \]

Since \( m_1 = -1 \) and \( m_2 = 1 \), their product is \(-1\), confirming that these sides are perpendicular.

Similar calculations for other pairs show all consecutive sides are perpendicular, confirming that this is a square.

3. Calculating Perimeter:

Perimeter \( P = 4 \times \text{side length} = 4 \times 3\sqrt{2} = 12\sqrt{2} \).

4. Calculating Area:

As it is a square, the area \( A = (\text{side length})^2 = (3\sqrt{2})^2 = 9 \times 2 = 18 \).

Summary for the first parallelogram:

- Type: Square

- Perimeter: \( 12\sqrt{2} \) units

- Area: 18 square units

Analysis of the Second Parallelogram

Vertices: (-1, 2), (4, 4), (2, -1), and (-3, -3).

1. Calculating Side Lengths:

- Between (-1, 2) and (4, 4):

\[ d = \sqrt{(4 + 1)^2 + (4 - 2)^2} = \sqrt{25 + 4} = \sqrt{29} \]

- Between (4, 4) and (2, -1):

\[ d = \sqrt{(2 - 4)^2 + (-1 - 4)^2} = \sqrt{4 + 25} = \sqrt{29} \]

- Between (2, -1) and (-3, -3):

\[ d = \sqrt{(-3 - 2)^2 + (-3 + 1)^2} = \sqrt{25 + 4} = \sqrt{29} \]

- Between (-3, -3) and (-1, 2):

\[ d = \sqrt{(-1 + 3)^2 + (2 + 3)^2} = \sqrt{4 + 25} = \sqrt{29} \]

All sides are of equal length (\(\sqrt{29}\)), indicating a rhombus.

2. Checking for Parallelogram and Rhombus:

To verify if it's a rhombus and parallelogram, confirm that pairs of opposite sides are parallel:

- Slope of (-1, 2) to (4, 4):

\[ m_1 = \frac{4 - 2}{4 + 1} = \frac{2}{5} \]

- Slope of (2, -1) to (-3, -3):

\[ m_2 = \frac{-3 + 1}{-3 - 2} = \frac{-2}{-5} = \frac{2}{5} \]

Vertices (-1, 2) to (4, 4) and (2, -1) to (-3, -3) are thus parallel; these are opposite sides.

- Slope of (4, 4) to (2, -1):

\[ m_3= \frac{-1 - 4}{2 - 4} = \frac{-5}{-2} = 2.5 \]

- Slope of (-3, -3) to (-1, 2):

\[ m_4= \frac{2 + 3}{-1 + 3} = \frac{5}{2} = 2.5 \]

Hence, both pairs confirm the figure as a rhombus and parallelogram with all sides equal and opposite sides parallel.

3. Calculating Perimeter:

\[ P = 4 \times \text{side length} = 4 \times \sqrt{29} \]

4. Calculating Area:

Using the shoelace formula:

\[

\text{Area} = \frac{1}{2} |x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1)|

\]

Plugging in:

\[

\begin{aligned}

& x_1=-1, y_1=2 \\

& x_2=4, y_2=4 \\

& x_3=2, y_3=-1 \\

& x_4=-3, y_4=-3

\end{aligned}

\]

Calculate:

\[

|(-1)(4) + 4(-1) + 2(-3) + (-3)(2) - (2)(4) + 4(2) + (-1)(-3) + (-3)(-1)| \\

= |-4 - 4 - 6 - 6 - (8 + 8 + 3 + 3)| \\

= |-20 - 22| = | -42 | = 42

\]

Area:

\[

\frac{1}{2} \times 42 = 21

\]

Summary for the second parallelogram:

- Type: Rhombus

- Perimeter: \( 4 \sqrt{29} \) units

- Area: 21 square units

Conclusion

The first parallelogram is classified as a square based on its equal sides and right angles, with a perimeter of \( 12\sqrt{2} \) units and an area of 18 square units. The second figure is a rhombus, characterized by equal sides and parallel opposite sides, with a perimeter of \( 4 \sqrt{29} \) units and an area of 21 square units. These analyses underscore the importance of coordinate geometry techniques—such as distance calculations, slope assessments, and the shoelace formula—in identifying the properties and dimensions of special quadrilaterals.

References

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