Problem 15 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (3 \sqrt{3}, \sqrt{5}) \text { and }(-\sqrt{3}, 4 \sqrt{5}) $$

Step-by-Step Solution

Answer

The distance between the points \((3\sqrt{3},\sqrt{5})\) and \((- \sqrt{3}, 4\sqrt{5})\) is \(\sqrt{93}\).

Step 1: Identify the coordinates

Identify the coordinates of the two given points on the plane. In this case, point A has coordinates \((x_1, y_1) = (3\sqrt{3}, \sqrt{5})\) and point B has coordinates \((x_2, y_2) = (-\sqrt{3}, 4\sqrt{5})\).

Step 2: Substitute into the distance formula

Substitute the coordinates of points A and B into the distance formula: \(d = \sqrt {(-\sqrt{3} - 3\sqrt{3})^2 + (4\sqrt{5} - \sqrt{5})^2}\). Simplifying under the square root: \(d = \sqrt{(-4\sqrt{3})^2 + (3\sqrt{5})^2}\).

Step 3: Simplify further

More simplification yields: \(d = \sqrt{4^2*\sqrt{3}^2 + 3^2*\sqrt{5}^2} = \sqrt{16*3 + 9*5} = \sqrt{48 + 45} = \sqrt{93}\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that mixes algebra with geometry to study and locate points, lines, and shapes on a coordinate plane. In a flat, two-dimensional plane, all points are determined by a pair of coordinates \(x, y\). These coordinates are used to position the point horizontally and vertically.

• The **x-coordinate** tells you how far to move horizontally from the origin, and the **y-coordinate** specifies the vertical position.
• Positive values mean right or up from the origin, while negative values imply moving left or down.
• The **origin**, represented by \(0, 0\), is the intersection point of the x-axis and y-axis.

Understanding coordinate geometry allows you to visualize and calculate properties of geometric figures, such as the distance between points. Let's use this to explore how we can find the distance between two points using a formula.

Distance Between Points

The primary method for finding the distance between two points in coordinate geometry is the Distance Formula. This mathematical tool takes the coordinates of two points and provides the straight-line distance between them. The Distance Formula is derived from the Pythagorean theorem, which can be expressed as:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here, \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are those of the second point.

• First, subtract the x-coordinates to find the horizontal distance.
• Next, subtract the y-coordinates for the vertical distance.
• Square each difference to ensure positivity and to apply the Pythagorean theorem.
• Finally, add these squared values and take the square root to find the distance.

For instance, if you have points A \(3\sqrt{3}, \sqrt{5}\) and B \(-\sqrt{3}, 4\sqrt{5}\), plug these into the formula to calculate the precise distance, as seen in the solution provided.

Simplifying Radicals

Simplifying radicals is an essential skill when working with the Distance Formula, as it often contains square roots. Simplifying helps to express the radicals in their simplest form to easily perform operations and understand the results. Here's a simple guide on how to simplify radicals:

• Factor the number under the square root into its prime factors.
• Separate pairs of identical factors because each pair can be moved outside the square root as a single number.
• If there are any numbers left without a pair, they stay under the square root.

Consider \(\sqrt{93}\) from the exercise. Since 93 does not have any perfect square factors other than 1, it is already in its simplest form. In the problem, radicals such as \(\sqrt{3}^2\) and \(\sqrt{5}^2\) were simplified by squaring to remove the radicals during the distance calculation.