Problem 49 - Introductory Algebra for College Students

Question

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(3,5) \text { and }(4,1)$$

Step-by-Step Solution

Answer

The distance between the points (3,5) and (4,1) is \(\sqrt{17}\) or approximately 4.12 when rounded to two decimal places.

Step 1: Identify the coordinates

The coordinates of the two points are P(3,5) and Q(4,1). We can identify \(x_1 = 3\), \(y_1 = 5\), \(x_2 = 4\), and \(y_2 = 1\).

Step 2: Substituting the coordinates into the distance formula

Substitute the coordinates into the formula \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This gives us \(D = \sqrt{(4 - 3)^2 + (1 - 5)^2}\).

Step 3: Simplifying the expression under the square root

Carry out the subtraction in the parenthesis and then square the result. This gives us \(D = \sqrt{(1)^2 + (-4)^2} = \sqrt{1 + 16}\).

Step 4: Further simplification and rounding off

Add the terms under the root to get \(D = \sqrt{17}\). This number cannot be simplified further, so we leave it in radical form. If necessary, this value can be rounded to two decimal places to provide an approximate value of \(D \approx 4.12\)

Key Concepts

Coordinates

When we talk about coordinates in the context of geometry or algebra, we're mainly referring to the system that allows us to specify the positions of points on a plane. The most common system used is the Cartesian coordinate system. Here, a point is represented by a pair of numbers, known as coordinates. For example, in the point (3,5), 3 represents the x-coordinate, and 5 represents the y-coordinate. These coordinates tell us exactly where the point sits relative to two perpendicular lines, or axes, known as the x-axis and y-axis.

• The first number in the pair, called the x-coordinate, tells us how far to move horizontally from the origin.
• The second number, known as the y-coordinate, tells us how far to move vertically.

Coordinates are essential as they provide a way to exactly describe the location of a point in space. Hence, understanding how to work with them is crucial for solving problems involving distance, lines, and shapes on planes.

Simplified Radical Form

When it comes to expressing numbers, especially distances in geometry, we often encounter radicals, also known as square roots. A number like \(\sqrt{17}\) is in what we call a simplified radical form. Simplified radical form means that the number is expressed as a root (or radical) in its simplest state. This implies there are no perfect square factors left in the number under the square root that can be pulled outside the square root sign.
For instance, if your expression was \(\sqrt{18}\), you could further simplify it because 18 is made up of a square number (9) and a non-square (2). So, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\). However, since 17 doesn’t have a perfect square factor except for 1, \(\sqrt{17}\) remains as is.
Understanding simplified radical form helps make complex math calculations more manageable and is often required for exact answers in various math problems.

Rounding to Decimal Places

Rounding numbers is an essential skill in mathematics, particularly when you require approximate answers or when the exact numbers are not suitable. Rounding to decimal places means adjusting a number to end at a certain number of decimal digits. For instance, rounding \(\sqrt{17} \approx 4.123\) to two decimal places, you would only keep two numbers after the decimal point, giving you approximately 4.12.
Rounding involves checking the number immediately after your desired decimal place. If that number is 5 or greater, you increase the last kept digit by 1. If it's less than 5, you leave the last kept digit the same.

• An important part of rounding is ensuring that you are consistent in your approach.
• Use the rounding technique based on the specific requirement, such as significant figures or decimal places.

In practical applications and exams, rounding off enables the handling of complex calculations with simpler numbers, while maintaining a reasonable level of precision.