Problem 14 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-\sqrt{2}) \text { and }(\sqrt{7}, 0) $$

Step-by-Step Solution

Answer

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

Step 1 - Identify the coordinates of both points

Here are our two points:Point 1 = (0, -\sqrt{2}) = (x_1, y_1)Point 2 = (\sqrt{7}, 0) = (x_2, y_2)

Step 2 - Plugging the coordinates into the distance formula

Substitute the coordinates into the formula. Follow the order (Point 2 - Point 1) for both x and y to avoid negative square roots. That gives us: d = \sqrt{{(\sqrt{7} - 0)^2 + (0 - (-\sqrt{2}))^2}}

Step 3 - Simplify the terms within the square root

Simplify the terms inside the square root. That gives us: d = \sqrt{{7 - 0^2 + 0 - (\sqrt{2})^2}}

Step 4 - Simplify further and Calculate the square root

d = \sqrt{{7 - 0 + 2}} = \sqrt{9}. Now, calculate the square root of 9, which is 3.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that relates to the sides of a right triangle. It states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, it is expressed as:\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. This theorem is not just limited to triangles on paper, but also very useful in a coordinate plane, where it helps find distances between points.When you have two points in a coordinate system, you can imagine a right triangle between them. The line connecting the points on the coordinate plane is the hypotenuse of the triangle. By applying the Pythagorean theorem, we can find the length of this line, or the distance between the two points.

Coordinate system

The coordinate system is a way of using numbers to represent points in space. In its simplest form, a coordinate plane is made up of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. Points are expressed as pairs of numbers (\(x, y\)), known as coordinates.

• The first number in the pair, \(x\), tells you how far to move right (positive) or left (negative) from the origin point (0,0).
• The second number, \(y\), tells you how far to move up (positive) or down (negative).

In the exercise example, point one is (0, -\(\sqrt{2}\)) and point two is (\(\sqrt{7}\), 0). These points position themselves on the coordinate plane at their respective coordinates. Understanding the coordinate system is essential for using the distance formula, which connects these points with a straight line to measure their distance.

Radicals

Radicals involve root operations, such as square roots, which simplify an expression under a radical sign. A square root, for example, is written as \(\sqrt{x}\) and represents the number that multiplies by itself to create \(x\). Radicals are crucial when dealing with the distance formula because we often end up needing to simplify expressions that include square roots.When working with radicals, it's key to remember some fundamental arithmetic rules which can make simplification easier:

• Simplifying \(\sqrt{x^2}\) gives \(x\).
• To add or subtract radicals, they must be like terms (have the same radicand).

In the distance formula from the exercise, we see radicals as part of the calculation. To find the distance between two points, we use the square root to simplify the sum of the squared differences of the coordinates. For example, the calculation of the distance \[ d = \sqrt{(\sqrt{7} - 0)^2 + (0 - (-\sqrt{2}))^2} \]is handled by effectively managing radical expressions which ultimately simplifies to find the actual distance.