Show That The Points ( − 6 , 4 (-6,4 ( − 6 , 4 ], ( − 2 , 2 (-2,2 ( − 2 , 2 ], And ( 4 , − 1 (4,-1 ( 4 , − 1 ] Are Collinear.

In geometry, three points are said to be collinear if they lie on the same straight line. In this article, we will show that the points $(-6,4)$, $(-2,2)$, and $(4,-1)$ are collinear.

What are Collinear Points?

Collinear points are points that lie on the same straight line. In other words, if three points are collinear, then they can be connected by a straight line. This means that the points will have the same slope or gradient.

The Concept of Slope

The slope of a line is a measure of how steep it is. It is calculated by dividing the vertical distance between two points by the horizontal distance between them. The slope of a line can be positive, negative, or zero.

Calculating the Slope

To show that the points $(-6,4)$, $(-2,2)$, and $(4,-1)$ are collinear, we need to calculate the slope between each pair of points.

Slope between $(-6,4)$ and $(-2,2)$

To calculate the slope between $(-6,4)$ and $(-2,2)$, we use the formula:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Plugging in the values, we get:

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

Slope between $(-2,2)$ and $(4,-1)$

To calculate the slope between $(-2,2)$ and $(4,-1)$, we use the same formula:

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

Plugging in the values, we get:

$$m = \frac{-1 - 2}{4 - (-2)} = \frac{-3}{6} = -\frac{1}{2}$$

Are the Points Collinear?

As we can see, the slope between $(-6,4)$ and $(-2,2)$ is $-\frac{1}{2}$, and the slope between $(-2,2)$ and $(4,-1)$ is also $-\frac{1}{2}$. This means that the points $(-6,4)$, $(-2,2)$, and $(4,-1)$ have the same slope, which is a necessary condition for them to be collinear.

Conclusion

In conclusion, we have shown that the points $(-6,4)$, $(-2,2)$, and $(4,-1)$ are collinear. This means that they lie on the same straight line and have the same slope.

Why is this Important?

Understanding collinear points is important in geometry and mathematics because it helps us to identify and work with straight lines. In many real-world applications, such as engineering, architecture, and computer graphics, we need to work with straight lines and understand how they relate to each other.

Real-World Applications

Collinear points have many real-world applications, such as:

• Engineering: In engineering, we use collinear points to design and build structures, such as bridges and buildings.
• Architecture: In architecture, we use collinear points to design and build buildings and other structures.
• Computer Graphics: In computer graphics, we use collinear points to create 3D models and animations.

Final Thoughts

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In our previous article, we showed that the points $(-6,4)$, $(-2,2)$, and $(4,-1)$ are collinear. In this article, we will answer some frequently asked questions about collinear points.

Q: What are collinear points?

A: Collinear points are points that lie on the same straight line. In other words, if three points are collinear, then they can be connected by a straight line.

Q: How do I determine if three points are collinear?

A: To determine if three points are collinear, you need to calculate the slope between each pair of points. If the slopes are the same, then the points are collinear.

Q: What is the significance of collinear points?

A: Collinear points are significant in geometry and mathematics because they help us to identify and work with straight lines. In many real-world applications, such as engineering, architecture, and computer graphics, we need to work with straight lines and understand how they relate to each other.

Q: Can three points be collinear if they are not on the same line?

A: No, three points cannot be collinear if they are not on the same line. Collinear points must lie on the same straight line.

Q: Can two points be collinear?

A: No, two points cannot be collinear. Collinear points must be three or more points that lie on the same straight line.

Q: Can a point be collinear with itself?

A: No, a point cannot be collinear with itself. Collinear points must be three or more distinct points that lie on the same straight line.

Q: Can a point be collinear with two other points if it is not on the same line?

A: No, a point cannot be collinear with two other points if it is not on the same line. Collinear points must lie on the same straight line.

Q: How do I find the equation of a line that passes through three collinear points?

A: To find the equation of a line that passes through three collinear points, you can use the point-slope form of a line. The point-slope form is given by:

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

where $(x_1, y_1)$ is one of the collinear points and $m$ is the slope of the line.

Q: Can I use the equation of a line to find the slope of a line that passes through three collinear points?

A: Yes, you can use the equation of a line to find the slope of a line that passes through three collinear points. The slope of the line is the coefficient of $x$ in the equation of the line.

Q: Can I use the equation of a line to find the equation of a line that passes through three collinear points?

A: Yes, you can use the equation of a line to find the equation of a line that passes through three collinear points. The equation of the line is given by:

$$y = mx + b$$

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

Q: Can I use the equation of a line to find the coordinates of a point that lies on a line that passes through three collinear points?

A: Yes, you can use the equation of a line to find the coordinates of a point that lies on a line that passes through three collinear points. The coordinates of the point are given by:

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

where $(x_1, y_1)$ is one of the collinear points and $m$ is the slope of the line.

Q: Can I use the equation of a line to find the distance between a point and a line that passes through three collinear points?

A: Yes, you can use the equation of a line to find the distance between a point and a line that passes through three collinear points. The distance between the point and the line is given by:

$$d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}$$

where $(x, y)$ is the point, $(A, B, C)$ is the equation of the line, and $d$ is the distance between the point and the line.

Conclusion

In conclusion, we have answered some frequently asked questions about collinear points. Collinear points are significant in geometry and mathematics because they help us to identify and work with straight lines. We hope that this article has been helpful in understanding collinear points and how to work with them.