Understanding the Relationship Between Segments FG and EH

Oct 13, 2025 by miodeya.com 17 views

Understanding the Relationship Between Segments FG and EH

In the world of mathematics, particularly in geometry and coordinate planes, understanding the relationships between different segments and lines is crucial. In this article, we will delve into the relationship between segments FG and EH, which are part of quadrilateral EFGH on a coordinate plane. We will explore how these segments are related and what statement proves this relationship.

Quadrilateral EFGH is on a coordinate plane. Segment FG is on the line $3x - y = -2$ , and segment EH is on the line $3x - y = -6$ . The problem asks us to find the statement that proves how segments FG and EH are related.

To understand the relationship between segments FG and EH, we need to analyze the lines on which they lie. The equation of a line in slope-intercept form is given by $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

The equation of the line on which segment FG lies is $3x - y = -2$ . We can rewrite this equation in slope-intercept form as $y = 3x + 2$ . This means that the slope of the line on which segment FG lies is $3$ .

Similarly, the equation of the line on which segment EH lies is $3x - y = -6$ . We can rewrite this equation in slope-intercept form as $y = 3x + 6$ . This means that the slope of the line on which segment EH lies is also $3$ .

Since the slopes of the lines on which segments FG and EH lie are the same, we can conclude that these segments are parallel. However, the problem asks us to find the statement that proves how segments FG and EH are related.

Option A: Opposite Reciprocal Slopes

Option A states that segments FG and EH have opposite reciprocal slopes. To understand this, let's first recall what opposite reciprocal slopes mean.

If two lines have slopes $m_1$ and $m_2$ , then their slopes are said to be opposite reciprocal if $m_1 \cdot m_2 = -1$ . In other words, if the product of the slopes of two lines is $-1$ , then their slopes are opposite reciprocal.

However, in this case, the slopes of the lines on which segments FG and EH lie are the same, which is $3$ . Therefore, the product of their slopes is $3 \cdot 3 = 9$ , not $-1$ . This means that the slopes of the lines on which segments FG and EH lie are not opposite reciprocal.

In conclusion, the statement that proves how segments FG and EH are related is not Option A. Since the slopes of the lines on which segments FG and EH lie are the same, we can conclude that these segments are parallel.

However, the problem asks us to find the statement that proves how segments FG and EH are related. Since the slopes of the lines on which segments FG and EH lie are the same, we can conclude that these segments are parallel. But we need to find the correct statement that proves this relationship.

The correct statement that proves how segments FG and EH are related is:

They are parallel.

This statement is true because the slopes of the lines on which segments FG and EH lie are the same, which is $3$ . Since the slopes of the lines on which segments FG and EH lie are the same, we can conclude that these segments are parallel.

In conclusion, the statement that proves how segments FG and EH are related is:

They are parallel.

The relationship between segments FG and EH can be understood by analyzing the lines on which they lie.
The slopes of the lines on which segments FG and EH lie are the same, which is $3$ .
Since the slopes of the lines on which segments FG and EH lie are the same, we can conclude that these segments are parallel.
The correct statement that proves how segments FG and EH are related is: They are parallel.
Quadrilateral EFGH: A Q&A Guide

In our previous article, we explored the relationship between segments FG and EH, which are part of quadrilateral EFGH on a coordinate plane. We concluded that these segments are parallel because the slopes of the lines on which they lie are the same. In this article, we will provide a Q&A guide to help you better understand the concepts and relationships involved.

Q: What is the relationship between segments FG and EH?

A: Segments FG and EH are parallel because the slopes of the lines on which they lie are the same.

Q: How do you determine if two lines are parallel?

A: To determine if two lines are parallel, you need to compare their slopes. If the slopes are the same, then the lines are parallel.

Q: What is the equation of a line in slope-intercept form?

A: The equation of a line in slope-intercept form is given by $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: How do you rewrite an equation in slope-intercept form?

A: To rewrite an equation in slope-intercept form, you need to isolate the variable $y$ on one side of the equation. This can be done by subtracting the product of the slope and the variable $x$ from both sides of the equation.

Q: What is the slope of the line on which segment FG lies?

A: The slope of the line on which segment FG lies is $3$ .

Q: What is the slope of the line on which segment EH lies?

A: The slope of the line on which segment EH lies is also $3$ .

Q: Why are segments FG and EH parallel?

A: Segments FG and EH are parallel because the slopes of the lines on which they lie are the same.

Q: What is the relationship between the slopes of two parallel lines?

A: The slopes of two parallel lines are the same.

Q: How do you determine if two lines are perpendicular?

A: To determine if two lines are perpendicular, you need to compare their slopes. If the product of the slopes is $-1$ , then the lines are perpendicular.

Q: What is the relationship between the slopes of two perpendicular lines?

A: The slopes of two perpendicular lines are opposite reciprocal.

Q: Can you provide an example of two lines that are perpendicular?

A: Yes, consider the lines $y = 2x + 1$ and $y = -\frac{1}{2}x - 2$ . The slopes of these lines are $2$ and $-\frac{1}{2}$ , respectively. Since the product of the slopes is $-1$ , these lines are perpendicular.

In conclusion, we hope this Q&A guide has helped you better understand the concepts and relationships involved in the relationship between segments FG and EH. Remember, the relationship between two lines can be understood by analyzing their slopes and equations.