Polygon $WXYZ$ Has Vertices $W(-1.5, 1.5$\], $X(6, 1.5$\], $Y(6, -4.5$\], And $Z(-1.5, -4.5$\]. Is $WXYZ$ A Rectangle? Justify Your Answer.

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Introduction

In geometry, a rectangle is a type of quadrilateral with four right angles and opposite sides of equal length. To determine if a given polygon is a rectangle, we need to check if it satisfies these properties. In this article, we will analyze the given polygon $WXYZ$ with vertices $W(-1.5, 1.5)$, $X(6, 1.5)$, $Y(6, -4.5)$, and $Z(-1.5, -4.5)$ to determine if it is a rectangle.

Properties of a Rectangle

A rectangle has the following properties:

• Opposite sides of equal length: The length of the opposite sides of a rectangle is equal.
• Four right angles: A rectangle has four right angles, which means that each internal angle is a right angle (90 degrees).
• Diagonals bisect each other: The diagonals of a rectangle bisect each other, meaning that they intersect at their midpoints.

Analysis of Polygon $WXYZ$

To determine if polygon $WXYZ$ is a rectangle, we need to check if it satisfies the properties of a rectangle.

Opposite Sides of Equal Length

To check if the opposite sides of polygon $WXYZ$ are of equal length, we need to calculate the distance between the vertices.

• Distance between $W$ and $X$: The distance between $W(-1.5, 1.5)$ and $X(6, 1.5)$ is given by:

  ${d_{WX} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

  ${d_{WX} = \sqrt{(6 - (-1.5))^2 + (1.5 - 1.5)^2}}$

  ${d_{WX} = \sqrt{(7.5)^2 + 0}}$

  ${d_{WX} = \sqrt{56.25}}$

  ${d_{WX} = 7.5}$

• Distance between $X$ and $Y$: The distance between $X(6, 1.5)$ and $Y(6, -4.5)$ is given by:

  ${d_{XY} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

  ${d_{XY} = \sqrt{(6 - 6)^2 + (-4.5 - 1.5)^2}}$

  ${d_{XY} = \sqrt{0 + (-6)^2}}$

  ${d_{XY} = \sqrt{36}}$

  ${d_{XY} = 6}$

• Distance between $Y$ and $Z$: The distance between $Y(6, -4.5)$ and $Z(-1.5, -4.5)$ is given by:

  ${d_{YZ} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

  ${d_{YZ} = \sqrt{(-1.5 - 6)^2 + (-4.5 - (-4.5))^2}}$

  ${d_{YZ} = \sqrt{(-7.5)^2 + 0}}$

  ${d_{YZ} = \sqrt{56.25}}$

  ${d_{YZ} = 7.5}$

• Distance between $Z$ and $W$: The distance between $Z(-1.5, -4.5)$ and $W(-1.5, 1.5)$ is given by:

  ${d_{ZW} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

  ${d_{ZW} = \sqrt{(-1.5 - (-1.5))^2 + (1.5 - (-4.5))^2}}$

  ${d_{ZW} = \sqrt{0 + (6)^2}}$

  ${d_{ZW} = \sqrt{36}}$

  ${d_{ZW} = 6}$

As we can see, the opposite sides of polygon $WXYZ$ are not of equal length. The distance between $WX$ and $YZ$ is 7.5, while the distance between $XY$ and $ZW$ is 6.

Four Right Angles

To check if polygon $WXYZ$ has four right angles, we need to calculate the internal angles.

• Internal angle at $W$: The internal angle at $W$ is given by:

  ${\angle W = \tan^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right)}$

  ${\angle W = \tan^{-1}\left(\frac{1.5 - 1.5}{6 - (-1.5)}\right)}$

  ${\angle W = \tan^{-1}\left(\frac{0}{7.5}\right)}$

  ${\angle W = \tan^{-1}(0)}$

  ${\angle W = 0}$

• Internal angle at $X$: The internal angle at $X$ is given by:

  ${\angle X = \tan^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right)}$

  ${\angle X = \tan^{-1}\left(\frac{-4.5 - 1.5}{6 - 6}\right)}$

  ${\angle X = \tan^{-1}\left(\frac{-6}{0}\right)}$

  ${\angle X = \tan^{-1}(-\infty)}$

  ${\angle X = 180}$

• Internal angle at $Y$: The internal angle at $Y$ is given by:

  ${\angle Y = \tan^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right)}$

  ${\angle Y = \tan^{-1}\left(\frac{-4.5 - (-4.5)}{6 - 6}\right)}$

  ${\angle Y = \tan^{-1}\left(\frac{0}{0}\right)}$

  ${\angle Y = \tan^{-1}(\infty)}$

  ${\angle Y = 90}$

• Internal angle at $Z$: The internal angle at $Z$ is given by:

  ${\angle Z = \tan^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right)}$

  ${\angle Z = \tan^{-1}\left(\frac{-4.5 - 1.5}{-1.5 - 6}\right)}$

  ${\angle Z = \tan^{-1}\left(\frac{-6}{-7.5}\right)}$

  ${\angle Z = \tan^{-1}\left(\frac{4}{5}\right)}$

  ${\angle Z = 36.87}$

As we can see, the internal angles at $W$ and $X$ are not right angles. The internal angle at $W$ is 0, while the internal angle at $X$ is 180.

Diagonals Bisect Each Other

To check if the diagonals of polygon $WXYZ$ bisect each other, we need to calculate the midpoint of each diagonal.

• Midpoint of $WX$: The midpoint of $WX$ is given by:

  ${M_{WX} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$

  ${M_{WX} = \left(\frac{-1.5 + 6}{2}, \frac{1.5 + 1.5}{2}\right)}$

  ${M_{WX} = \left(\frac{4.5}{2}, \frac{3}{2}\right)}$

  ${M_{WX} = \left(2.25, 1.5\right)}$

• Midpoint of $XY$: The midpoint of $XY$ is given by:

  ${M_{XY} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$

  ${M_{XY} = \left(\frac{6 + 6}{2}, \frac{1.5 + (-4.5)}{2}\right)}$

  ${M_{XY} = \left(\frac{12}{2}, \frac{-3}{2}\right)}$

  ${M_{XY} = \left(6, -1.5\right)}$

• Midpoint of $YZ$: The midpoint of $YZ$ is given by:

  ${M_{YZ} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$

  ${M_{YZ} = \left(\frac{-1.5 + 6}{2}, \frac{-4.5 + (-4.5)}{2}\right)}$

  [M_{YZ} = \left(\frac{
  Q&A: Is Polygon $WXYZ$ a Rectangle? =====================================

Q: What is a rectangle?

A: A rectangle is a type of quadrilateral with four right angles and opposite sides of equal length.

Q: What are the properties of a rectangle?

A: A rectangle has the following properties:

• Opposite sides of equal length: The length of the opposite sides of a rectangle is equal.
• Four right angles: A rectangle has four right angles, which means that each internal angle is a right angle (90 degrees).
• Diagonals bisect each other: The diagonals of a rectangle bisect each other, meaning that they intersect at their midpoints.

Q: How do we determine if a polygon is a rectangle?

A: To determine if a polygon is a rectangle, we need to check if it satisfies the properties of a rectangle. We can do this by calculating the distance between the vertices, the internal angles, and the midpoints of the diagonals.

Q: What is the distance between the vertices of polygon $WXYZ$?

A: The distance between the vertices of polygon $WXYZ$ is as follows:

• Distance between $W$ and $X$: 7.5
• Distance between $X$ and $Y$: 6
• Distance between $Y$ and $Z$: 7.5
• Distance between $Z$ and $W$: 6

As we can see, the opposite sides of polygon $WXYZ$ are not of equal length.

Q: What are the internal angles of polygon $WXYZ$?

A: The internal angles of polygon $WXYZ$ are as follows:

• Internal angle at $W$: 0
• Internal angle at $X$: 180
• Internal angle at $Y$: 90
• Internal angle at $Z$: 36.87

As we can see, the internal angles at $W$ and $X$ are not right angles.

Q: What are the midpoints of the diagonals of polygon $WXYZ$?

A: The midpoints of the diagonals of polygon $WXYZ$ are as follows:

• Midpoint of $WX$: (2.25, 1.5)
• Midpoint of $XY$: (6, -1.5)
• Midpoint of $YZ$: (-1.5, -4.5)

As we can see, the midpoints of the diagonals do not intersect at the same point.

Q: Is polygon $WXYZ$ a rectangle?

A: No, polygon $WXYZ$ is not a rectangle. It does not satisfy the properties of a rectangle.

Q: Why is polygon $WXYZ$ not a rectangle?

A: Polygon $WXYZ$ is not a rectangle because it does not have opposite sides of equal length, four right angles, and diagonals that bisect each other.

Q: What type of polygon is $WXYZ$?

A: Polygon $WXYZ$ is a quadrilateral with four sides and four vertices.

Q: Can we determine the type of polygon $WXYZ$ is?

A: Yes, we can determine the type of polygon $WXYZ$ is by analyzing its properties. Based on the calculations, we can see that polygon $WXYZ$ is a quadrilateral with four sides and four vertices.

Q: What are the possible types of quadrilaterals?

A: The possible types of quadrilaterals are:

• Rectangle: A quadrilateral with four right angles and opposite sides of equal length.
• Square: A quadrilateral with four right angles, opposite sides of equal length, and all sides of equal length.
• Rhombus: A quadrilateral with four sides of equal length and opposite sides of equal length.
• Trapezoid: A quadrilateral with four sides and two pairs of opposite sides of equal length.

Q: Which type of quadrilateral is polygon $WXYZ$?

A: Based on the calculations, we can see that polygon $WXYZ$ is a trapezoid. It has four sides and two pairs of opposite sides of equal length.

Q: Why is polygon $WXYZ$ a trapezoid?

A: Polygon $WXYZ$ is a trapezoid because it has four sides and two pairs of opposite sides of equal length. The opposite sides are $WX$ and $YZ$, and the other pair of opposite sides are $XY$ and $ZW$.