Problem 4 - 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. $$(2,-3) \text { and }(-1,5)$$

Step-by-Step Solution

Answer

The distance between the pair of points is approximately 8.54.

Step 1: Identify the coordinates of the points

Let the first point (2, -3) be represented by coordinates (x1, y1) and the second point (-1, 5) be represented by coordinates (x2, y2). So, x1 = 2, y1 = -3, x2 = -1, y2 = 5.

Step 2: Apply the distance formula

After identifying the coordinates, the calculation of the distance d between the two points can proceed using the formula \[d = \sqrt{(x2-x1)^2+(y2-y1)^2}\]. Plugging in the numbers: \[d = \sqrt{(-1-2)^2+(5-(-3))^2} = \sqrt{(-3)^2+(8)^2} = \sqrt{9+64}\]

Step 3: Simplify the result

Continue with simplifying the result within the square root \[d = \sqrt{73}\].

Step 4: Round to two decimal places

Finally, round the result to the nearest two decimal places. This gives \(d \approx 8.54\)

Key Concepts

Coordinate Geometry

Coordinate Geometry is a fascinating branch of mathematics that involves the study of geometric figures using a coordinate system. It merges algebra with geometry to help us analyze shapes and determine measurements, such as distance and area, on a plane. Understanding coordinate geometry is essential for dealing with points and lines efficiently.

In coordinate geometry, we describe points using ordered pairs. Points like \(2, -3\) or \(-1, 5\) lie on a flat plane called the Cartesian plane. This plane is defined by a horizontal x-axis and a vertical y-axis. Each axis helps define a point's position in terms of distance from a reference point known as the origin \(0, 0\).

Points in a Plane

A plane is a two-dimensional surface where we can plot different points using coordinates. Each point on this plane is defined by two values: the x-coordinate and the y-coordinate.

The x-coordinate shows how far left or right the point is from the y-axis, while the y-coordinate tells us how high or low the point is relative to the x-axis.

• The point \(2, -3\) means starting at the origin, move 2 units to the right along the x-axis, and then 3 units down from the x-axis.
• For \(-1, 5\), you move 1 unit to the left and 5 units up.

Understanding how to navigate these points helps us use formulas like the distance formula to find distances between them. It’s like a map that tells us exactly where every point is located.

Simplified Radical Form

Simplifying expressions is an important algebraic skill, and in geometry, we often need to simplify square roots. When we calculate distances using the distance formula, we sometimes end up with complex square roots that can be daunting at first glance.

When you find a distance, like \(\sqrt{73}\) between two points, simplifying helps make the number easier to understand and use. Sometimes the simplest form is the only step before converting it to a decimal approximation, such as rounding \(\sqrt{73}\) to **8.54**.

• Simplify radical expressions by finding and factoring out perfect squares.
• Use a calculator for decimal approximations when necessary, but keep in simplified form if the task asks for exact results.

Simplifying radicals is not only about computation; it also teaches clean, concise mathematical writing.