Question

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(2,-3) ; \quad P_{2}=(4,2) $$

Short answer

\( d = \sqrt{29} \)

Step-by-step solution

- Identify the coordinates

First, note the coordinates of the points given: Point 1 (\(P_{1}\)) has coordinates (2, -3).Point 2 (\(P_{2}\)) has coordinates (4, 2).

- Recall the distance formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

- Substitute the coordinates into the formula

Substitute the coordinates of \(P_{1}\) and \(P_{2}\) into the distance formula: \[ d = \sqrt{(4 - 2)^2 + (2 - (-3))^2} \]

- Simplify inside the square root

Simplify the expressions inside the square root: \[ d = \sqrt{(2)^2 + (5)^2} \]

- Calculate the squares

Calculate the squares of the numbers: \[ d = \sqrt{4 + 25} \]

- Add the results

Add the results inside the square root: \[ d = \sqrt{29} \]

- Simplify the square root (if possible)

The square root of 29 cannot be simplified further. Thus, \[ d = \sqrt{29} \]

Key concepts

distance between points

One of the core concepts in coordinate geometry is finding the distance between points. Understanding this concept is crucial for solving various problems on a coordinate plane. The distance formula is derived from the Pythagorean theorem. If you have two points \((P_{1})\) and \((P_{2})\) with coordinates \((x_1, y_1)\) and \((x_2, y_2)\), you can find the distance between them using the distance formula.
The formula is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula calculates the length of the hypotenuse of a right triangle formed by the horizontal and vertical distances between the two points.
Always start by identifying the coordinates of the points. Label \((x_1, y_1)\) and \((x_2, y_2)\) correctly to avoid confusion.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows you to solve geometric problems algebraically and to understand spatial relationships better.
In coordinate geometry, every point in the plane is defined by a pair of numbers called coordinates. The horizontal coordinate is called the x-coordinate, and the vertical coordinate is called the y-coordinate.
When finding the distance between points, you work within this coordinate system. You use the x-coordinates and y-coordinates to determine the distance horizontally and vertically. By applying the distance formula, you connect algebraic representations with geometric interpretations.

• Always plot your points on a graph if possible.
• Double-check your coordinates.

Coordinate geometry combines algebra and geometry, providing tools to solve problems involving distances, midpoints, slopes, and more.

distance calculation in algebra

Distance calculation in algebra involves using algebraic methods to find the distance between two points in a coordinate plane. The step-by-step method we saw in the original exercise is a typical example:

• Identify the coordinates of the given points.
• Recall and write down the distance formula.
• Substitute the coordinates into the formula.
• Simplify the expression inside the square root.
• Perform the arithmetic operations to find the result.

This procedure turns a geometric problem into an algebraic one, making it easier to handle with basic arithmetic and algebra skills.
Remember, the key is to substitute the coordinates correctly and perform arithmetic operations step-by-step. This ensures accuracy in your calculations.
Practice these steps regularly to become proficient in finding distances in algebra.