Problem 1 - 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. $$(2,3) \text { and }(14,8)$$

Step-by-Step Solution

Answer

The distance between the points (2,3) and (14,8) is 13.

Step 1: Prepare for calculation

First, identify the given coordinates: let (2,3) be (x1, y1) and (14,8) be (x2, y2).

Step 2: Substitute the values into the formula

Substitute x1=2, y1=3, x2=14, y2=8 into the distance formula: \(\sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}\).

Step 3: Complete the operation

After simplifying, you would get: \(\sqrt{(14-2)^2 + (8 - 3)^2} = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169}\).

Step 4: Find the Root

Taking the square root of 169, we get 13 as our answer. This is the exact answer in integer form.

Step 5: Expressing in Decimal

As 13 doesn't require any simplification or rounding, it will remain unchanged as the exact distance between the two points.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of geometry where we use a pair of coordinates to represent points on a plane. These coordinates are like a map that tells us the exact location of a point using two numbers: an x-coordinate and a y-coordinate. The x-coordinate tells us how far the point is from the vertical axis (also known as the y-axis), while the y-coordinate tells us how far the point is from the horizontal axis (also known as the x-axis).

When analyzing points, such as (2,3) and (14,8), it helps to understand that we're essentially plotting these points on a two-dimensional grid. The first number is always the x-coordinate, and the second number is the y-coordinate. This system allows us to visualize mathematical operations and measurements, like distance, much more clearly. By utilizing these tools from Coordinate Geometry, determining how far apart these points are becomes a matter of applying the right formula.

Distance Calculation

Distance Calculation in coordinate geometry involves using the Distance Formula. This formula helps us calculate the straight-line distance between two points in a coordinate plane. This is particularly useful when you need the shortest possible path from point A to point B.

To find the distance, you substitute the coordinates into the Distance Formula:

• Let \[(x_1, y_1)\] be the first point and \[(x_2, y_2)\] be the second point.
• The formula is \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
• Plug in the specific values, simplify the expression inside the square root, and finally find the square root to determine the distance.

In our example, the distance between points \((2,3)\) and \((14,8)\) was found to be \(13\), which is the exact distance on a straight line between them. This simple yet powerful technique helps us not only understand how far two locations are from each other but also demonstrates the elegance of mathematics in real-world applications.

Radical Expressions

Radical Expressions occur when you are dealing with roots, specifically square roots, in mathematical operations. These are expressions that include a radical symbol \(\sqrt{}\). In geometry, this often arises when using the Distance Formula, as seen in our example.

When calculating the distance, we reached the step \(\sqrt{169}\). This is a radical expression where the radicand (the number under the square root sign) is \(169\). Calculating a square root is the process of finding a number which, when multiplied by itself, results in the original number. Here, \(13\cdot13 = 169\), hence the square root \(\sqrt{169}\) is \(13\).

When working with radical expressions, it's important to simplify them as much as possible. However, some results (like \(\sqrt{169}\)) may already be in their simplest form. Knowing how to deal with these expressions makes it easier to handle various geometry problems and ensures accurate calculations, especially when needing an exact value.