Problem 15 - 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. $$(3 \sqrt{3}, \sqrt{5}) \text { and }(-\sqrt{3}, 4 \sqrt{5})$$

Step-by-Step Solution

Answer

The distance between the points \((3 \sqrt{3}, \sqrt{5})\) and \((-\sqrt{3}, 4 \sqrt{5})\) is \(\sqrt{93}\) units, which approximately equals 9.64 units.

Step 1: Identifying the points

Identify the given pair of points. Here, the pair of points are \((3 \sqrt{3}, \sqrt{5})\) and \((-\sqrt{3}, 4 \sqrt{5})\). Therefore, \(x_{1}=3 \sqrt{3}\), \(y_{1}=\sqrt{5}\), \(x_{2}=-\sqrt{3}\), and \(y_{2}=4 \sqrt{5}\).

Step 2: Substituting in the distance formula

Substitute the values of \(x_{1}\), \(y_{1}\), \(x_{2}\), and \(y_{2}\) into the distance formula. Therefore, Distance = \(\sqrt{{(-\sqrt{3}-3 \sqrt{3})}^{2} + {(4 \sqrt{5}-\sqrt{5})}^{2}}\).

Step 3: Simplify the expression

Simplify the expression. The expression reduces to Distance = \(\sqrt{{(-4 \sqrt{3})}^{2} + {(3 \sqrt{5})}^{2}}\). This further simplifies to Distance = \(\sqrt{48+45}\). This again simplifies to Distance = \(\sqrt{93}\).

Step 4: Convert to decimal form and round off

Convert \(\sqrt{93}\) to decimal form using a calculator and round off to two decimal places. The Distance = 9.64

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric figures using a coordinate system. This typically involves using a pair of numerical coordinates to represent points on a plane. Points are located using an ordered pair, \( (x, y) \) in the Cartesian coordinate system, which consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

Using coordinate geometry, we can solve problems related to distances, midpoints, gradients, and equations of lines. For instance, to find the distance between two points in a plane, the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) is used, wherein \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. This formula is derived from the Pythagorean theorem, which applies to right triangles formed by drawing vertical and horizontal lines through the given points.

Radical Expressions

Radical expressions involve the use of roots, such as square roots, cube roots etc. In the context of the distance formula, we usually come across square roots, as we are dealing with distances in two dimensions. The square root of a number \( x \) is written as \( \sqrt{x} \) and represents a value that, when multiplied by itself, gives \( x \).

Radical expressions often need to be simplified for ease of understanding and computation. Simplifying involves identifying and factoring out perfect squares from the radicand, which is the number or expression beneath the radical sign. For instance, \( \sqrt{93} \) cannot be simplified further as 93 is not a product of a perfect square and another integer. However, if we had \( \sqrt{48} \) like in our solution, this simplifies to \( \sqrt{16 \times 3} \) which further simplifies to \( 4\sqrt{3} \) since 16 is a perfect square.

Decimal Approximation

Decimal approximation is the process of expressing numbers in a more manageable form when an exact figure is not required, or cannot easily be obtained. This is particularly useful when dealing with irrational numbers or radical expressions, whose exact decimal representation may be infinite or very long.

For example, the distance between two points given as \( \sqrt{93} \) is an irrational number. Since we cannot represent the square root of 93 exactly as a fraction, we approximate it to a decimal. Calculators can help find this decimal approximation, which can then be rounded to a desired degree of accuracy, commonly to two decimal places for practical applications. For our exercise, the decimal approximation of \( \sqrt{93} \) is approximately 9.64 when rounded to two decimal places. This approximation is convenient for measurements, such as when calculating distances, because exact numbers are not always necessary or practical.