Zandolph Is Creating Rectangle WXYZ Such That WX Has An Equation Of Y = 1 4 X + 4 Y=\frac{1}{4}x+4 Y = 4 1 ​ X + 4 . Segment XY Must Pass Through The Point ( − 2 , 6 (-2, 6 ( − 2 , 6 ]. Which Of The Following Is The Equation For XY?A. Y − 6 = 1 4 ( X − ( − 2 ) Y - 6 = \frac{1}{4}(x - (-2) Y − 6 = 4 1 ​ ( X − ( − 2 ) ]B.

Introduction

In mathematics, particularly in geometry and algebra, we often encounter problems that involve finding the equation of a line that passes through a given point and has a specific slope. In this article, we will explore how to find the equation of segment XY in rectangle WXYZ, given that WX has an equation of $y=\frac{1}{4}x+4$ and segment XY must pass through the point $(-2, 6)$.

Understanding the Problem

To solve this problem, we need to understand the concept of slope and how it relates to the equation of a line. The slope of a line is a measure of how steep it is, and it can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $m$ is the slope, and $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

In this case, we are given the equation of line WX, which is $y=\frac{1}{4}x+4$. This means that the slope of line WX is $\frac{1}{4}$.

Finding the Equation of XY

Now, we need to find the equation of segment XY, which must pass through the point $(-2, 6)$. To do this, we can use the point-slope form of a line, which is:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

In this case, we know that the point $(-2, 6)$ lies on segment XY, and we also know that the slope of line WX is $\frac{1}{4}$. However, we need to find the slope of segment XY.

Calculating the Slope of XY

To find the slope of segment XY, we can use the fact that the slope of a line is the same at any point on the line. Since we know the equation of line WX, we can use it to find the slope of segment XY.

Let's call the point where segment XY intersects line WX as point A. We can use the equation of line WX to find the coordinates of point A.

Finding the Coordinates of Point A

To find the coordinates of point A, we can substitute the x-coordinate of point A into the equation of line WX and solve for the y-coordinate.

Let's call the x-coordinate of point A as $x_A$. Then, we can substitute $x_A$ into the equation of line WX as follows:

$$y_A = \frac{1}{4}x_A + 4$$

Now, we need to find the value of $x_A$. To do this, we can use the fact that point A lies on segment XY, which means that the x-coordinate of point A is the same as the x-coordinate of point $(-2, 6)$.

Finding the Value of $x_A$

Since point A lies on segment XY, we know that the x-coordinate of point A is the same as the x-coordinate of point $(-2, 6)$. Therefore, we can set $x_A = -2$.

Now, we can substitute $x_A = -2$ into the equation of line WX to find the y-coordinate of point A:

$$y_A = \frac{1}{4}(-2) + 4$$

Simplifying the equation, we get:

$$y_A = -\frac{1}{2} + 4$$

$$y_A = \frac{7}{2}$$

Therefore, the coordinates of point A are $(-2, \frac{7}{2})$.

Finding the Slope of XY

Now that we have the coordinates of point A, we can use them to find the slope of segment XY.

The slope of segment XY is the same as the slope of line WX at point A. Therefore, we can use the equation of line WX to find the slope of segment XY:

$$m_{XY} = \frac{1}{4}$$

Finding the Equation of XY

Now that we have the slope of segment XY, we can use it to find the equation of segment XY.

We can use the point-slope form of a line to find the equation of segment XY:

$$y - 6 = \frac{1}{4}(x - (-2))$$

Simplifying the equation, we get:

$$y - 6 = \frac{1}{4}(x + 2)$$

$$y - 6 = \frac{1}{4}x + \frac{1}{2}$$

$$y = \frac{1}{4}x + \frac{25}{2}$$

Therefore, the equation of segment XY is $y = \frac{1}{4}x + \frac{25}{2}$.

Conclusion

In this article, we have shown how to find the equation of segment XY in rectangle WXYZ, given that WX has an equation of $y=\frac{1}{4}x+4$ and segment XY must pass through the point $(-2, 6)$. We have used the point-slope form of a line and the equation of line WX to find the slope of segment XY, and then used it to find the equation of segment XY.

Answer

The equation of segment XY is $y = \frac{1}{4}x + \frac{25}{2}$.

References

• [1] "Point-Slope Form of a Line" by Math Open Reference
• [2] "Slope of a Line" by Math Is Fun
• [3] "Equation of a Line" by Khan Academy
  Frequently Asked Questions (FAQs) about Finding the Equation of XY ====================================================================

Q: What is the point-slope form of a line?

A: The point-slope form of a line is a way to write the equation of a line using the slope of the line and a point on the line. It is given by the equation:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the equation of line WX?

A: The equation of line WX is given by:

$$y = \frac{1}{4}x + 4$$

Q: What is the point $(-2, 6)$?

A: The point $(-2, 6)$ is a point on segment XY.

Q: How do I find the equation of segment XY?

A: To find the equation of segment XY, you can use the point-slope form of a line and the equation of line WX. First, find the slope of segment XY using the equation of line WX. Then, use the point-slope form of a line to find the equation of segment XY.

Q: What is the equation of segment XY?

A: The equation of segment XY is given by:

$$y = \frac{1}{4}x + \frac{25}{2}$$

Q: How do I know if the equation of segment XY is correct?

A: To check if the equation of segment XY is correct, you can substitute the coordinates of a point on segment XY into the equation and see if it is true.

Q: What if I don't know the equation of line WX?

A: If you don't know the equation of line WX, you can use the slope-intercept form of a line to find it. The slope-intercept form of a line is given by:

$$y = mx + b$$

where $m$ is the slope, and $b$ is the y-intercept.

Q: Can I use the equation of segment XY to find the equation of line WX?

A: No, you cannot use the equation of segment XY to find the equation of line WX. The equation of segment XY is a separate equation that is used to describe the relationship between the x and y coordinates of points on segment XY.

Q: What if I have a different equation for line WX?

A: If you have a different equation for line WX, you can use it to find the equation of segment XY. However, you will need to use the point-slope form of a line and the new equation for line WX to find the equation of segment XY.

Q: Can I use the equation of segment XY to find the equation of a different line?

A: No, you cannot use the equation of segment XY to find the equation of a different line. The equation of segment XY is a specific equation that is used to describe the relationship between the x and y coordinates of points on segment XY, and it cannot be used to find the equation of a different line.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of XY. We have covered topics such as the point-slope form of a line, finding the slope of a line, and using the equation of line WX to find the equation of segment XY. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the topic.