Problem 12 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (3,-2),(0,3) $$

Step-by-Step Solution

Answer

The distance between points (3,-2) and (0,3) is approximately 5.83.

Step 1: Identifying the coordinates

Given points are (3, -2) and (0, 3), here (x1, y1) represents the first point and (x2, y2) represents the second point. So, x1 = 3, y1 = -2, x2 = 0, y2 = 3.

Step 2: Substitute the identified coordinates in the distance formula

Solve for \(\sqrt{(x2-x1)^2 + (y2-y1)^2}\) by substituting the identified coordinates into the formula which then becomes \(\sqrt{(0-3)^2 + (3-(-2))^2}\.

Step 3: Calculate the distance

Calculate the above expression, \(\sqrt{(0-3)^2 + (3-(-2))^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9+25} = \sqrt{34}\). The distance is \(\sqrt{34}\) or approximately 5.83 when rounded to the nearest hundredth.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves algebra and geometry. It allows us to find the position of points in a plane using a coordinate system based on a pair of numerical coordinates. These coordinates, usually represented as

• x-coordinate (horizontal position)
• y-coordinate (vertical position)

are crucial for various calculations, including distances, slopes, and midpoints.
In our exercise, we have coordinates (3, -2) and (0, 3). Each pair represents a different point in a two-dimensional plane. By understanding their positions, we can further explore the relationships and calculations possible, such as finding the distance between these two points.

Distance Between Points

To find the distance between two points in a coordinate plane, we use the distance formula. This formula is derived from the Pythagorean theorem, which applies to right-angled triangles. For any two points

• (x1, y1)
• (x2, y2)

the distance between them, denoted as d, can be calculated using the formula: \[d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\]This formula helps in finding the direct line distance, also known as Euclidean distance, between the two points. By plugging in our points (3, -2) and (0, 3), we find the distance: \[= \sqrt{(0-3)^2 + (3-(-2))^2} = \sqrt{9 + 25} = \sqrt{34}\]This method reliably gives us a precise measurement of how far apart the points are, essential for tasks involving space or distance.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify complex numbers, making them easier to work with while maintaining an acceptable level of accuracy. When calculating distances or other precise measures, we often encounter non-whole numbers.
For instance, in our exercise, the distance calculated is \(\sqrt{34}\) which is approximately 5.830951...
To make it easier to interpret, we round this number to the nearest hundredth, resulting in 5.83. This tells us that the distance between the points is effectively 5.83 units apart.

• Rounding follows certain rules, primarily rounding up if the next digit is five or greater.
• It makes numeric data cleaner and more intuitive, especially when conveying results succinctly.

Understanding when and why to round numbers is crucial in scenarios that require numerical presentation for reports or everyday use.