Problem 14 - 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. $$(0,-\sqrt{2}) \text { and }(\sqrt{7}, 0)$$

Step-by-Step Solution

Answer

Therefore, the distance between the points \((0,-\sqrt{2})\) and \((\sqrt{7}, 0)\) is \(3\).

Step 1: Identifying Coordinates for Each Point

Identify the coordinates for Point A and Point B. For Point A, the coordinates are \(0\) and \(-\sqrt{2}\), and for Point B, the coordinates are \(\sqrt{7}\) and \(0\).

Step 2: Adding Coordinates into the Distance Formula

Put the coordinates into the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). When substituting it will look like this: \(d = \sqrt{(\sqrt{7} - 0)^2 + (0 - (-\sqrt{2}))^2}\) = \(\sqrt{7 + 2}\). This is the simplified radical form.

Step 3: Computing the Distance

Now, compute the distance by calculating the square root of \(9\), which results in \(3\).

Key Concepts

Understanding Coordinates

Coordinates help us locate points on a plane. Think of them as the address of a point. In your home address, you have a street number and city. Similarly, each point on a plane has two numbers, an \(x\)-value and a \(y\)-value. These numbers tell us how far along and how far up the point is located from the origin (where \(x = 0\) and \(y = 0\)).

For example, given the points \((0, -\sqrt{2})\) and \((\sqrt{7}, 0)\):

• For the point \((0, -\sqrt{2})\), the \(x\)-value is 0 and the \(y\)-value is \(-\sqrt{2}\).
• For the point \((\sqrt{7}, 0)\), the \(x\)-value is \(\sqrt{7}\) and the \(y\)-value is 0.

Identifying these coordinates is the first step when finding the distance between two points.

Expressing in Simplified Radical Form

The simplified radical form is a neat way to express a square root without decimals. It helps keep solutions clean and precise. Once we have the values from the distance formula, simplifying these values into radical form ensures that we deal with exact numbers rather than estimates.

Looking at our solution process:

• We use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
• Substitute the points: \(d = \sqrt{(\sqrt{7} - 0)^2 + (0 - (-\sqrt{2}))^2}\).
• This simplifies to \(d = \sqrt{7 + 2}\), which can be expressed as \(\sqrt{9}\).

Even though \(\sqrt{9}\) simplifies further to a nice integer value, understanding how to express these numbers in radical form is the key step here.

Performing Square Root Calculations

Calculating the square root is a fundamental skill in math. When you see a square root symbol, you're finding what number multiplies by itself to give you the original number under the square root.

In our example:

• We need to find \(\sqrt{9}\).
• Think about which number, when multiplied by itself, equals 9. That number is 3.

Performing this calculation might seem simple, but it's important for solving many math problems. Understanding square roots helps in finding lengths, distances, and other quantities in geometry.