Problem 3 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (4,-1) \text { and }(-6,3) $$

Step-by-Step Solution

Answer

The approximate distance between the points (4,-1) and (-6,3) is 10.77

Step 1: Identify the coordinates

The coordinates for the first point are (4,-1). Let these be \(x1=4\) and \(y1=-1\). The coordinates for the second point are (-6,3). Let these be \(x2=-6\) and \(y2=3\).

Step 2: Apply the distance formula

Substitute the coordinates of the two points into the distance formula: \(d=\sqrt{(-6-4)^2+(3-(-1))^2}\)

Step 3: Solve the equation

First calculate the squares inside the parentheses and then add the two results: \(d=\sqrt{(-10)^2+(4)^2} = \sqrt{100+16} = \sqrt{116}\)

Step 4: Simplify the expression

The root of 116 can be rounded to two decimal places, giving an approximate distance of \(d = 10.77\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that deals with geometric shapes using a coordinate system. This system allows for precise definitions and calculations. In the Cartesian coordinate system, each point on a plane is defined by an ordered pair of numbers, which are usually represented by

• An x-coordinate, which determines the point's horizontal position.
• A y-coordinate, which determines the point's vertical position.

Given a point with coordinates (x, y), you can easily locate its position on a grid.
This coordinate system provides a bridge between algebra and geometry, making it possible to calculate distances, slopes, and midpoints between points efficiently. Understanding how to plot and interpret points on the Cartesian plane is fundamental in solving geometry problems.
In contexts such as navigating or casting a location onto a map, mastering coordinate geometry is key to understanding the spatial relations between different objects.

Calculating Distance Between Points

One of the essential uses of coordinate geometry is calculating the distance between two points on a plane. The distance formula, derived from the Pythagorean theorem, is used for this purpose. If you have two points,

• Point 1: (x_1, y_1)
• Point 2: (x_2, y_2),

the distance \(d\) between them can be calculated using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This equation allows us to find the straight-line distance between two points. In our example, the points are (4, -1) and (-6, 3). By plugging in these values, we compute:\[-6 - 4 = -10\]\[3 - (-1) = 4\]Squaring these differences and adding them gives:\[(-10)^2 + (4)^2 = 100 + 16 = 116\]Taking the square root of 116 will yield the distance.

Rounding Numbers

Rounding is a skill used to simplify numbers by reducing the digits while keeping the value close to what it originally was. This is particularly useful when dealing with numbers that are too precise, like those resulting from square roots.
Different rounding rules fulfill different purposes:

• Rounding down truncates the numbers.
• Rounding up increases the smallest place value that we need.
• Rounding to a certain decimal place discards all remaining decimal points beyond the set precision.

In math problems involving measurements, rounding to two decimal places is common. To do this, you view the third decimal place.
If it is 5 or more, round the second decimal place up; if it is less than 5, round down. In our earlier example, \(\sqrt{116}\) has more decimals than needed.
We round it to get a final distance of 10.77, which is easier to manage and communicate.