Question

Plot the given points in the coordinate plane and then find the distance between them. $$ (-3,5),(2,-2) $$

Short answer

The distance between (-3, 5) and (2, -2) is \( \sqrt{74} \).

Step-by-step solution

Understanding the Task

The task requires you to first plot the given points on the coordinate plane, and then to calculate the distance between these two points using the distance formula. We will achieve this step-by-step.

Identify the Points

The points given are \((-3, 5)\) and \((2, -2)\). These points need to be plotted on a coordinate plane.

Plotting the Points

To plot the points, locate each point on the coordinate plane by considering the first number as the x-coordinate and the second as the y-coordinate. Mark the point \((-3, 5)\) 3 units left and 5 units up of the origin, and the point \((2, -2)\) 2 units right and 2 units down from the origin.

Understanding the Distance Formula

The distance formula to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] We will use this formula to find the distance between the points.

Substitute the Values

For the points \((-3, 5)\) and \((2, -2)\), substitute the values into the formula as follows: \[ d = \sqrt{(2 - (-3))^2 + (-2 - 5)^2} \] Simplify inside the parentheses first.

Calculate Inside Parentheses

Evaluate the expressions inside the parentheses: \[ (2 - (-3)) = 2 + 3 = 5 \]\[ (-2 - 5) = -7 \] Now substitute these values back into the formula.

Calculate Squared Values

Square the two results:\[ 5^2 = 25 \]\[ (-7)^2 = 49 \] Add these squared values together to continue with the distance formula.

Sum and Find Square Root

Add the squared values: \[ 25 + 49 = 74 \]Finally, take the square root of this sum to find the distance: \[ d = \sqrt{74} \] This is the final distance between the two points.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where points are plotted based on their coordinates, which are pairs of numerical values. These values represent the point's position in the plane.
The plane is formed by two perpendicular number lines intersecting at their zero points, called the origin, marked by \(0,0\). The horizontal line is the x-axis, and the vertical line is the y-axis.
This grid-like system allows us to precisely position any point using ordered pairs known as coordinates, where the first value denotes the x-axis position and the second value denotes the y-axis position.

• X-value: Determines how far left or right the point is from the origin.
• Y-value: Determines how far up or down the point is from the origin.

This simple yet powerful concept is fundamental in geometry and helps visually represent mathematical problems, such as finding the distance between two points.

Distance Formula

The distance formula is a mathematical equation derived from the Pythagorean theorem. It calculates the straight line distance between two points on a coordinate plane.
For points \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \(d\) is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula helps us determine the length of the hypotenuse of a right triangle formed by projecting the line segment onto the x-axis and y-axis.

• \(x_2 - x_1\): The horizontal distance between points.
• \(y_2 - y_1\): The vertical distance between points.
• Squaring the distances: Ensures all values are positive and accommodates for direction changes in the plane.
• Square root: Returns the direct distance by reversing the squaring process.

This method can be easily used to calculate how far apart two locations are without complex calculations.

Plotting Points

Plotting points on a coordinate plane involves marking the location of a point using its coordinates. This allows for a visual understanding of the data or relationship between points.
To plot a point, start by identifying its coordinates, represented as \( (x, y) \). Then, follow these steps:

• Find the x-coordinate: Move right if positive, left if negative, from the origin.
• Find the y-coordinate: Move up if positive, down if negative, from your x-position.
• Mark the point: Place a dot where the x and y-coordinates intersect.

For example, to plot \((-3, 5)\), move three units left from the origin and five units up.
Similarly, for \(2, -2)\), move two units right and two units down. Once both points are plotted, they can be connected to visualize their relationship, aiding in understanding tasks such as calculating distances or slopes.