Problem 19 - College Algebra with Corequisite Support

Question

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers. (5,0) and (5,6)

Step-by-Step Solution

Answer

The distance between the points is 6 units.

Step 1: Understand the Distance Formula

The distance formula to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] This formula is derived from the Pythagorean theorem.

Step 2: Identify the Coordinates

For the points given in the exercise, identify the coordinates:- Point 1: \((5,0)\);- Point 2: \((5,6)\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics, especially useful in geometry and algebra. It connects the sides of a right triangle, allowing us to find lengths easily. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In formula terms, it is written as: \[c^2 = a^2 + b^2\]where \(c\) represents the hypotenuse, and \(a\) and \(b\) are the other two sides.This theorem is particularly useful not just for solving triangles, but also when applied in the coordinate plane. It lays the foundation for the Distance Formula, which helps us find the length between two points in a plane. By thinking of the change in x and y coordinates as the two sides of a right triangle, the Pythagorean Theorem helps us calculate that straight-line distance, which is essentially the hypotenuse.

Coordinate Geometry

Coordinate Geometry, sometimes called analytical geometry, is about studying geometry using a coordinate system. In simpler terms, it’s about understanding geometry through algebraic equations and how they relate to graphical figures.Key aspects of Coordinate Geometry include:

• Points: Specified by coordinates (x, y) in the Cartesian plane.
• Distance Formula: Used to calculate the straight-line distance between two points.
• Midpoint Formula: Finds the middle point between two given points.
• Slope Formula: Determines the steepness or inclination of a line connecting two points.

When using the Distance Formula, you imagine the line connecting two points on the coordinate plane as the hypotenuse of a right triangle. The changes in x values (\(x_2 - x_1\)) and y values (\(y_2 - y_1\)) are akin to the other two sides of the triangle. This connection makes Coordinate Geometry a powerful tool for interpreting graphical data or problems in mathematics.

Radical Expressions

Radical expressions are mathematical expressions that contain a root symbol, often referred to as a radical. The most common radical is the square root (\( \sqrt{} \)), but cubic roots and other forms also exist.Understanding how radical expressions work is essential, as they frequently appear in equations, particularly those involving the Distance Formula. Simplifying radical expressions means expressing them in their simplest form. This process often involves:

• Breaking them down into smaller prime factors.
• Combining like terms when applicable.
• Rationalizing denominators to remove radicals, if necessary.

For example, consider simplifying \( \sqrt{36} \). Because 36 is a perfect square, it is simply \( 6 \). However, more complex radicals, such as \( \sqrt{50} \), simplify to \( 5\sqrt{2} \). Simplifying radical expressions is vital in providing the exact answer, as seen in the distance calculations from the original exercise where we express distances in simplest radical form.