Problem 16 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact distance between the points (2√3, √6) and (-√3, 5√6) is \(\sqrt{123}\). Rounded to two decimal places, the distance is approximately 11.09.

Step 1: Identify the points

We have two points: the first is \((2\sqrt{3}, \sqrt{6})\), and the second is \((- \sqrt{3}, 5 \sqrt{6})\). We'll call the first point (x1, y1) and the second point (x2, y2).

Step 2: Apply the distance formula

The formula for the distance between two points (x1, y1) and (x2, y2) in a coordinate system is \(\sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}\). Substituting the given points into the formula gives: \(\sqrt{((- \sqrt{3} - 2\sqrt{3})^{2} + (5\sqrt{6} - \sqrt{6})^{2})}\)

Step 3: Simplify the expression

We can simplify the expression above as: \(\sqrt{((-3\sqrt{3})^{2} + (4\sqrt{6})^{2})}\), which further simplifies to \(\sqrt{(27 + 96)}\). Hence the exact distance in simplified radical form is \(\sqrt{123}\)

Step 4: Round to two decimal places

If we want the numerical approximation of the distance to two decimal places, we just compute the square root of 123 which is approximately 11.09.

Key Concepts

Coordinate Geometry

Coordinate geometry is a crucial concept in mathematics that involves studying points, lines, and shapes in the coordinate plane. It combines algebra and geometry to provide a comprehensive understanding of geometric figures using coordinates.

• Each point on the plane is represented by a pair of numbers, known as coordinates, usually written as \((x, y)\).
• The x-coordinate indicates the point's horizontal position, while the y-coordinate represents its vertical position.

In the exercise, we are given two points with coordinates involving square roots: \((2\sqrt{3}, \sqrt{6})\) and \(-\sqrt{3}, 5\sqrt{6})\). These coordinates allow us to calculate the distance between the two points using a specific formula known as the distance formula. This formula is derived from the Pythagorean theorem and applies in any coordinate plane.

Simplified Radical Form

Expressing answers in a simplified radical form means reducing a square root expression to its simplest form. This is important in mathematics, particularly when dealing with distances in coordinate geometry. It makes expressions neat and manageable.

• Simplification reduces complexity by factoring the number under the square root into prime factors and finding perfect squares.
• For example, if you start with \(\sqrt{123}\), you check if any factors can be written as squares.

In the original exercise, we discovered that\(\sqrt{123}\)could not be simplified further because 123 has no perfect square factors larger than 1. Thus, it stays in its existing radical form. Simplifying expressions as much as possible helps with both exact and approximate solutions.

Square Root Calculation

Calculating the square root is a fundamental part of solving distance problems in coordinate geometry. The square root represents a value that, when multiplied by itself, results in the original number.

• The square root symbol (\(\sqrt{\ }\)) denotes this operation, and finding the square root means determining such a value.
• In the exercise, after simplifying the expression to \(\sqrt{123}\), the square root calculation provides the final distance value.

To approximate the distance to two decimal places, we calculate\(\sqrt{123}\)approximately using a calculator, which gives us about 11.09. Understanding square root calculation is vital for solving practical problems involving distances and dimensions.