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
Related questions
Question
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 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d0c88f8-9b0e-4a87-ae6c-39dd67e2599b%2F3e2ea99e-2433-411e-822f-b5943249f255%2Fwxvmlv_processed.jpeg&w=3840&q=75)
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d0c88f8-9b0e-4a87-ae6c-39dd67e2599b%2F3e2ea99e-2433-411e-822f-b5943249f255%2Fsddtu0v_processed.jpeg&w=3840&q=75)
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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