Problem 16 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (2 \sqrt{3}, \sqrt{6}) \text { and }(-\sqrt{3}, 5 \sqrt{6}) $$

Step-by-Step Solution

Answer

The distance between points (2√3, √6) and (-√3, 5√6) is \( \sqrt {123} \).

Step 1: Identify coordinates

Identify the coordinates from the given question. Here, point A: (2√3, √6) and point B: (-√3, 5√6).

Step 2: Insert into formula

Insert the coordinates into the distance formula. Distance \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} = \sqrt{((-√3 - 2√3)^2 + (5√6 - √6)^2)}\).

Step 3: Simplify inside the square roots

Simplify the expressions inside the square root: = \( \sqrt{((-3√3)^2 + 4√6^2)} = \sqrt{(27 + 96)} \).

Step 4: Continue to simplify

Continue simplification to get the distance: = \( \sqrt{123} \).

Key Concepts

Coordinate Geometry

Coordinate geometry (also known as analytic geometry) is a mathematical tool used to study geometry using an algebraic approach. This involves using a coordinate system to describe geometric figures.

The most common coordinate system used in this case is the Cartesian coordinate system, which uses two perpendicular lines (usually called the x-axis and y-axis) that intersect at a point known as the origin. Each point in this plane can be described using a pair of numbers, known as coordinates, which correspond to distances from the two axes.

In the problem given, we are tasked with finding the distance between two points on a coordinate plane: \((2\sqrt{3}, \sqrt{6})\) and \((-\sqrt{3}, 5\sqrt{6})\). To do this, we use the distance formula, which derives from the Pythagorean Theorem and tells us the straight-line distance between two points.

Square Root Simplification

Simplification of square roots often appears in coordinate geometry problems, mainly because distance calculations involve finding the square root of a sum of squares. When you encounter square roots, it's important to simplify them as much as possible to work with expressions more easily.

The simplification process involves resolving the expression inside the square root to its simplest form. Look for any perfect squares that can be factored out of the expression within the radicand (the number inside the square root).

In our exercise, the expressions inside the roots are simplified as follows: \((-3\sqrt{3})^2 + (4\sqrt{6})^2\). Here, each term arises from subtracting the x and y coordinates of the two points respectively. Simplifying these terms individually, then, adding them gives \(27 + 96\), which when added produces the square root simplification \(\sqrt{123}\). Each of these steps reduces the complexity of the expression, making it easier to calculate the result.

Distance Calculation

The distance calculation between two points in coordinate geometry relies on a specific formula. This formula, known as the distance formula, is derived from the Pythagorean theorem. It allows us to calculate the distance as if the line segment connecting the two points forms the hypotenuse of a right triangle.

The formula is expressed as:

• \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

By identifying the coordinates of points A and B from the exercise, we substitute them into the formula.

This step-by-step substitution involves:

• Calculating \((x_2-x_1)^2\) and \((y_2-y_1)^2\).
• Adding these results together.
• Taking the square root of this sum.

After simplifying inside the square root, as shown, we find that the distance between the given points is \(\sqrt{123}\). This process ensures that we can calculate the precise distance between any two points efficiently.