Problem 18 - College Algebra

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. $$(2,-5) \text { and }(7,4)$$

Step-by-Step Solution

Answer

The distance between the points is \(\sqrt{106}\).

Step 1: Identify the coordinates of the points

The two points given in the exercise are \((x_1, y_1) = (2, -5)\) and \((x_2, y_2) = (7, 4)\).

Step 2: Use the distance formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

Step 3: Substitute the values into distance formula

Substitute the values of \((x_1, y_1)\) and \((x_2, y_2)\) into the distance formula:\[ d = \sqrt{(7 - 2)^2 + (4 - (-5))^2} \].

Step 4: Calculate the differences

Calculate the differences: \((x_2 - x_1) = 7 - 2 = 5\), and \((y_2 - y_1) = 4 - (-5) = 4 + 5 = 9\).

Step 5: Square the differences

Square the differences:\((x_2 - x_1)^2 = 5^2 = 25\), and \((y_2 - y_1)^2 = 9^2 = 81\).

Step 6: Add the squared differences

Add the squared differences:\[ 25 + 81 = 106 \].

Step 7: Compute the square root

Take the square root of the sum:\[ d = \sqrt{106} \].

Step 8: Simplify the radical expression

The expression \(\sqrt{106}\) is already in its simplest radical form, as 106 does not have any perfect square factors other than 1.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a fundamental aspect of mathematics that deals with points, lines, and figures expressed using coordinates in a plane. It provides a powerful visual representation of concepts that can often be abstract.

• A coordinate plane consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, denoted as (0,0).
• Coordinates are written as ordered pairs (x, y) where x indicates the position along the horizontal axis and y denotes the position along the vertical axis.
• In problems involving geometry, such as finding the distance between points, coordinates help in converting geometric ideas into algebraic expressions, making complex calculations more manageable.

For example, to find the distance between two points, we need only their coordinates, and from these, we can use the Distance Formula.

Radical Expressions

Radical Expressions involve numbers or expressions under a root, often the square root. Understanding these expressions is crucial for simplifying and manipulating them within mathematical calculations.

• The square root of a number is a value that, when multiplied by itself, gives the original number.
• Radical expressions can include integers, fractions, or variables under the root.
• When simplifying radical expressions, to find the simplest form, look for perfect square factors under the square root.

In coordinate geometry, especially when using the Distance Formula, we often end up with a square root, which must be simplified to get the most reduced expression, usually asked for in its simplest radical form.

Simplifying Radicals

Simplifying Radicals is the process of manipulating a radical expression to its simplest form. This involves breaking down the number under the root into its prime factors and simplifying any perfect squares.

• First, factor the number under the square root into its prime components.
• Then, pair the primes into groups of two because the square root of a perfect square is a whole number.
• If no pairs or other factors come out, the radical is already in its simplest form.

For instance, in our exercise, \(\sqrt{106}\) is already simplified because 106 does not have a perfect square factor other than 1. Understanding how to simplify radicals is crucial not only in geometry but also in algebra and calculus, where accurate expression and simplification of answers matter.