# Coordinate Geometry: Distance Formula and Slope of a Line Quiz

InvulnerableBeryllium
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## Coordinate Geometry: Distance Formula and Slope of a Line

In the realm of coordinate geometry, two fundamental concepts — distance formula and slope of a line — unlock the power to measure and analyze the behavior of points and lines on a plane. These ideas are based on the Cartesian coordinate system, where each point is defined by its (x)- and (y)-coordinates. Let's take a closer look at these two essential concepts.

### Distance Formula

The distance formula is a fundamental tool in coordinate geometry that enables us to find the distance between any two points ((x_1, y_1)) and ((x_2, y_2)) on a coordinate plane.

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Here, (d) is the distance between the two points. This formula is based on the Pythagorean theorem, which states that the length of the hypotenuse of a right-angled triangle is the square root of the sum of the squares of the lengths of the other two sides.

### Slope of a Line

The slope of a line is a measure of its steepness, giving us an indication of how much the line rises or falls in the (y)-direction for each unit increase in the (x)-direction. The slope (m) of a line that passes through points ((x_1, y_1)) and ((x_2, y_2)) is defined as follows:

[ m = \frac{y_2 - y_1}{x_2 - x_1} ]

If the line is horizontal (steepness is 0), then the slope is 0, and the equation of the line has the form (y = k) for some constant (k). If the line is vertical (steepness is undefined), then the slope is undefined, and the equation has the form (x = k) for some constant (k).

### Applications

The distance formula and slope of a line have numerous applications in coordinate geometry, including:

1. Calculating the distance between any two points on a coordinate plane.
2. Determining whether or not two lines intersect, and if they do, finding the point of intersection.
3. Locating the midpoint of a line segment connecting two points.
4. Finding the equation of a line passing through two points.
5. Solving equations involving lines and line segments.

### Example Problems

1. Find the distance between the points ((2, 4)) and ((5, 7)).

Using the distance formula, we have:

[ d = \sqrt{(5 - 2)^2 + (7 - 4)^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \text{ units} ]

1. Find the slope of the line passing through the points ((-1, 2)) and ((3, 7)).

Using the slope formula, we have:

[ m = \frac{7 - 2}{3 - (-1)} = \frac{5}{4} ]

### Conclusion

The distance formula and slope of a line are two fundamental concepts in coordinate geometry that enable us to measure and analyze the behavior of points and lines on a plane. These ideas are essential in graphing, solving equations, and understanding the relationships between points and lines. By mastering these concepts, you will have a strong foundation for exploring the rich world of coordinate geometry.

## Description

Explore fundamental concepts in coordinate geometry such as the distance formula and slope of a line, which are essential for measuring and analyzing points and lines on a plane. Learn how to calculate distances between points and determine the steepness of lines based on their coordinates. Mastering these concepts will provide a strong foundation for understanding relationships between points and lines in a Cartesian coordinate system.