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What is the coordinate matrix of that describes the following quadrilateral ABCD?

What is the coordinate matrix of that describes the following quadrilateral ABCD?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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What is the coordinate matrix of that describes the following quadrilateral ABCD?

### Title: Understanding Parallelograms in Coordinate Geometry #### Overview This lesson covers the properties and representation of parallelograms on a coordinate plane. Parallelograms are four-sided figures with opposite sides that are both equal and parallel. #### Diagram Description The diagram above is a graph featuring a parallelogram plotted on a coordinate plane. The vertices of the parallelogram are labeled as \(A\), \(B\), \(C\), and \(D\). ##### Details of the Diagram: - **Vertices:** - \( A \) is positioned at (5, 0) - \( B \) located at (6, 5) - \( C \) situated at (-4, 1), and - \( D \) at (-5, -4) - **Axis:** - The horizontal axis is the x-axis, marked with values. - The vertical axis is the y-axis, also marked with values. - **Plot Points:** - Each vertex is marked with a blue dot for clear visibility. - **Shape:** - The shape enclosed by connecting these vertices forms a parallelogram. #### Mathematical Properties 1. **Parallel Sides:** - \( AB \parallel CD \) - \( AD \parallel BC \) 2. **Equal Opposite Sides:** - \( AB = CD \) - \( AD = BC \) #### Understanding Coordinate Geometry: - **Slope Calculation:** - The slope of \(AB\) and the slope of \(CD\) would be equal, indicating that these lines are parallel. - Similarly, the slope of \(AD\) and \(BC\) would also be equal. - **Distance Formula:** - To confirm the lengths of the sides are equal, you can use the distance formula between the respective points: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] #### Practical Exercise: - **Finding the Midpoint:** - As an exercise, students can find the midpoint of each diagonal \(AC\) and \(BD\) to verify they bisect each other. ```markdown The midpoint formula is: M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
Transcribed Image Text:### Title: Understanding Parallelograms in Coordinate Geometry #### Overview This lesson covers the properties and representation of parallelograms on a coordinate plane. Parallelograms are four-sided figures with opposite sides that are both equal and parallel. #### Diagram Description The diagram above is a graph featuring a parallelogram plotted on a coordinate plane. The vertices of the parallelogram are labeled as \(A\), \(B\), \(C\), and \(D\). ##### Details of the Diagram: - **Vertices:** - \( A \) is positioned at (5, 0) - \( B \) located at (6, 5) - \( C \) situated at (-4, 1), and - \( D \) at (-5, -4) - **Axis:** - The horizontal axis is the x-axis, marked with values. - The vertical axis is the y-axis, also marked with values. - **Plot Points:** - Each vertex is marked with a blue dot for clear visibility. - **Shape:** - The shape enclosed by connecting these vertices forms a parallelogram. #### Mathematical Properties 1. **Parallel Sides:** - \( AB \parallel CD \) - \( AD \parallel BC \) 2. **Equal Opposite Sides:** - \( AB = CD \) - \( AD = BC \) #### Understanding Coordinate Geometry: - **Slope Calculation:** - The slope of \(AB\) and the slope of \(CD\) would be equal, indicating that these lines are parallel. - Similarly, the slope of \(AD\) and \(BC\) would also be equal. - **Distance Formula:** - To confirm the lengths of the sides are equal, you can use the distance formula between the respective points: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] #### Practical Exercise: - **Finding the Midpoint:** - As an exercise, students can find the midpoint of each diagonal \(AC\) and \(BD\) to verify they bisect each other. ```markdown The midpoint formula is: M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
### Linear Algebra: Matrix Options The image presents a multiple-choice question with four different matrices. Each option is accompanied by a circle to select the matrix as an answer. Below are the details of each matrix option: 1. **Option A:** \[ \begin{pmatrix} 5 & 8 & -1 & -4 \\ 2 & 3 & 3 & 2 \end{pmatrix} \] 2. **Option B:** \[ \begin{pmatrix} 7 & 10 & -1 & -6 \\ 3 & 4 & 4 & 3 \end{pmatrix} \] 3. **Option C:** \[ \begin{pmatrix} 5 & 8 & -1 & -4 \\ -1 & -6 & 6 & 1 \end{pmatrix} \] 4. **Option D:** \[ \begin{pmatrix} 5 & 8 & -1 & -4 \\ 1 & 6 & 6 & 1 \end{pmatrix} \] These matrices could be part of a larger question pertaining to operations like matrix multiplication, determinants, or identifying certain properties. Be sure to carefully choose the correct option based on the given problem.
Transcribed Image Text:### Linear Algebra: Matrix Options The image presents a multiple-choice question with four different matrices. Each option is accompanied by a circle to select the matrix as an answer. Below are the details of each matrix option: 1. **Option A:** \[ \begin{pmatrix} 5 & 8 & -1 & -4 \\ 2 & 3 & 3 & 2 \end{pmatrix} \] 2. **Option B:** \[ \begin{pmatrix} 7 & 10 & -1 & -6 \\ 3 & 4 & 4 & 3 \end{pmatrix} \] 3. **Option C:** \[ \begin{pmatrix} 5 & 8 & -1 & -4 \\ -1 & -6 & 6 & 1 \end{pmatrix} \] 4. **Option D:** \[ \begin{pmatrix} 5 & 8 & -1 & -4 \\ 1 & 6 & 6 & 1 \end{pmatrix} \] These matrices could be part of a larger question pertaining to operations like matrix multiplication, determinants, or identifying certain properties. Be sure to carefully choose the correct option based on the given problem.
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