Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((2,-5),(-6,1)\)

Short answer

The distance between the points \( (2,-5) \) and \( (-6,1) \) is approximately 9.22 units and the midpoint of the line segment joining these points is \( (-2,-2) \).

Step-by-step solution

Plotting the Points

Plot the points \( (2,-5) \) and \( (-6,1) \) on the Cartesian coordinate system. The first number in each pair is the x-coordinate and the second number is the y-coordinate.

Compute Distance Between Points

Use the Distance formula to calculate the distance between these two points, which is \( d = \sqrt{((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2})} \). Here, \( (x_{1},y_{1})=(2,-5) \) and \( (x_{2},y_{2})=(-6,1) \). Substituting the given values into the formula, compute the distance.

Determine the Midpoint

The Midpoint formula is used to find the midpoint of the line segment, which is \( M = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}) \). Here, \( (x_{1},y_{1}) = (2,-5) \) and \( (x_{2},y_{2}) = (-6,1) \). By plugging those values into the formula, calculate the midpoint.

Key concepts

Distance Formula

The Distance Formula is a handy tool in Analytic Geometry, useful for finding the length between two points on a plane. It stems from the Pythagorean theorem, making it perfect for this purpose. To apply the Distance Formula, you need two points, let's say \((x_1, y_1)\) and \((x_2, y_2)\). The formula is written as: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Here's how it works step by step:

• First, subtract the x-coordinates: \(x_2-x_1\).
• Next, subtract the y-coordinates: \(y_2-y_1\).
• Square both results to avoid any negative values.
• Add these squared differences together.
• Finally, take the square root of this sum to get the distance.

This formula helps determine the straight line or "as-the-crow-flies" distance between the two specified locations in the Cartesian coordinate system.

Midpoint Formula

Finding the midpoint is equally essential in Analytic Geometry. The midpoint is simply the point that is exactly halfway along the line segment connecting two points. It is calculated using the Midpoint Formula which is written as: \[M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\]To use the Midpoint Formula, keep these steps in mind:

• Add the x-coordinates of the two points: \(x_1 + x_2\).
• Add the y-coordinates: \(y_1 + y_2\).
• Divide each of these sums by 2 to get the average.

The result gives you the x and y coordinates of the midpoint, which lies directly between your two points on the plane. This concept is particularly useful when you're trying to find the center between two objects, such as cities on a map or stars in the sky.

Cartesian Coordinate System

The Cartesian Coordinate System is foundational in the study of Analytic Geometry. It's a two-dimensional plane formed by two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical). This system allows us to pinpoint locations using coordinates formatted as \((x, y)\). In this set-up, any point on the plane can be identified based on how far it is along the x and y axes from the origin, which is the point \((0,0)\). Here are some key concepts:

• The x-coordinate tells you how far left or right the point is.
• The y-coordinate tells you how far up or down the point is.
• An ordered pair \((x, y)\) is used to describe each point's location on the grid.
• The four quadrants of the plane help in determining the sign of the coordinates, making it easier to plot points accurately.

The Cartesian Coordinate System is crucial because it enables us to perform various calculations and geometric constructions with precision, opening up a wide range of possibilities in both math and science.