Problem 17 - 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. $$\left(\frac{7}{3}, \frac{1}{5}\right) \text { and }\left(\frac{1}{3}, \frac{6}{5}\right)$$

Step-by-Step Solution

Answer

The distance between the two points is approximately 2.24.

Step 1: Identify the Points

Firstly, identify the points. The two points given are \(\left(\frac{7}{3}, \frac{1}{5}\right)\) and \(\left(\frac{1}{3},\frac{6}{5}\right)\)

Step 2: Apply the Distance Formula

Apply the distance formula which is derived from the Pythagorean theorem. It is expressed as : \(d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\). In this case, \(x_1 = \frac{7}{3},\ y_1 = \frac{1}{5},\ x_2 = \frac{1}{3} \text{, and } y_2 = \frac{6}{5}\)

Step 3: Substitute the Values

Substitute the given values into the formula: \(d= \sqrt{{(\frac{1}{3}-\frac{7}{3})}^2 + {(\frac{6}{5}-\frac{1}{5})}^2}\)

Step 4: Simplify the Expression

Simplify to find the distance: \(d =\sqrt{{(-2)}^2+1} = \sqrt{4+1} = \sqrt{5}\)

Step 5: Round Off to Two Decimal Places

Finally, round off \(\sqrt{5}\) to two decimal places: \(d \approx 2.24\)

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, specifically pertaining to right triangles. It asserts that in a right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, this can be expressed as: \( c^2 = a^2 + b^2 \) where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

When determining the distance between two points in a coordinate plane, we essentially form a right triangle with the horizontal and vertical distances between the points as the legs, and the line segment joining them as the hypotenuse. Thus, the Pythagorean theorem becomes incredibly useful when utilizing the distance formula in coordinate geometry.

Radical Simplification

Radical simplification involves the process of simplifying expressions that contain roots. A common radical that often requires simplification is the square root. To simplify a square root, one looks for factors that are perfect squares and can be taken out of the radical. For example, \( \sqrt{8} \) can be simplified to \( 2\sqrt{2} \) because 8 is equal to 4 multiplied by 2, and 4 is a perfect square.

This process is especially important when working with the distance formula in precalculus. After calculating the sum of the squares of the differences of the x- and y-coordinates, the resulting value under the square root may need to be simplified to express the answer in a more understandable form, as we strive to find the most basic expression for the distance between two points.

Decimal Approximation

Decimal approximation is a method used to express numbers that are difficult to represent exactly, such as irrational numbers, in a simpler and more usable form. This is done by rounding the number to a certain number of decimal places based on the desired level of precision.

For instance, in the exercise provided, the distance calculated using the distance formula resulted in \( \sqrt{5} \), which is an irrational number. It cannot be precisely represented as a finite or repeating decimal. In practical terms, we often need to convert such values into decimal form, rounding them off to a certain number of decimal places for ease of use and understanding. As in the solution, \( \sqrt{5} \) was approximated to 2.24, which rounds the exact value to two decimal places.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This field enables the precise analysis and representation of geometric figures and the relationships between them by translating shapes and points into algebraic equations involving coordinates.

In the context of precalculus problems, when finding the distance between two points, we use a rectangular coordinate system to plot these points. Here, each point is specified by an ordered pair of numbers: \( (x, y) \) that represent horizontal and vertical distances from the origin. The distance formula, derived from the Pythagorean theorem, is a powerful tool in coordinate geometry that allows us to calculate the exact distance between any two points plotted in this system. It reflects the application of geometric principles to algebraic representations, unifying these two fundamental branches of mathematics.