Problem 20 - Algebra 1

Question

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(4,5),(-1,3)$$

Step-by-Step Solution

Answer

The distance between the points (4,5) and (-1,3) is approximately 5.39 units.

Step 1: Identify the coordinates

The problem gives two points in a plane, (4,5) and (-1,3), where (4,5) is the point \((x1, y1)\) and (-1,3) is the point \((x2, y2)\).

Step 2: Substitute into the distance formula

Substitute these points into the distance formula: \(d = \sqrt{((-1) - 4)^2 + (3 - 5)^2}\).

Step 3: Calculate the difference

First, find the differences and their squares: \(d = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4}\).

Step 4: Calculate the sqrt

Next, add the squares and then find the square root to get the distance: \(d = \sqrt{29} = 5.39\) (rounded to the nearest hundredth).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the branch of mathematics that studies geometric properties and relationships by using coordinate systems. In the context of this exercise, we are working with a two-dimensional coordinate plane that consists of an x-axis and a y-axis. Each point in this plane is represented by a pair of coordinates \(x, y\). For instance, the points \(4, 5\) and \(-1, 3\) are plotted based on their x and y values. Here are some important points about coordinate geometry:

• The x-coordinate indicates the position of a point relative to the vertical y-axis.
• The y-coordinate indicates the position of a point relative to the horizontal x-axis.
• The distance between two points can be calculated using the distance formula, which is crucial in many applications of coordinate geometry.

Understanding how to represent points and use formulas within the coordinate plane is fundamental to solving problems related to distance and many other geometric inquiries.

Square Root Calculation

Square root calculation is an essential part of using the distance formula in coordinate geometry. Given an expression like \( oot 29\), which stems from our exercise, finding the square root is crucial. Here's how we calculate it:

• First, the terms under the square root are summed up, such as \(25 + 4 = 29\).
• The square root operation \(\sqrt{}\) essentially asks what number, when multiplied by itself, equals the given number under the root.
• In this example, we use a calculator to find that the square root of 29 is approximately 5.385.

Calculating square roots often involves using a calculator, especially when working with non-perfect squares like 29. This calculation provides the actual distance between points in the coordinate plane calculated using the distance formula.

Rounding Decimals

Rounding decimals is a straightforward yet essential skill in mathematics, particularly when dealing with results from calculations such as square roots. In our exercise, we needed to round the calculated distance of \(5.385\) to the nearest hundredth. Here's how:

• First, identify the digit in the hundredths place, which is the second digit after the decimal point (in this case, 8).
• Next, observe the digit in the thousandth's place, which is the digit immediately after the hundredths place (in this case, 5).
• If this digit is 5 or greater, increase the hundredths place by 1. If it is less than 5, leave the hundredths place as it is.

By following the steps, we round \(5.385\) to \(5.39\), providing a cleaner and more manageable number. Rounding is critical for estimation and ensuring results are presented in a practical format that aligns with precision standards.