Problem 16 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ \left(\frac{7}{3}, \frac{1}{5}\right) \text { and }\left(\frac{1}{3}, \frac{6}{5}\right) $$

Step-by-Step Solution

Answer

The distance between the points \((\frac{7}{3}, \frac{1}{5})\) and \((\frac{1}{3}, \frac{6}{5})\) is approximately \(2.24\).

Step 1: Identify the coordinates

The first point is \((\frac{7}{3}, \frac{1}{5})\) and the second point is \((\frac{1}{3}, \frac{6}{5})\). Therefore, \(x_1=\frac{7}{3}\), \(y_1=\frac{1}{5}\), \(x_2=\frac{1}{3}\), and \(y_2=\frac{6}{5}\).

Step 2: Substitute the coordinates into the distance formula

The formula is \(\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}\). Substituting the coordinates into the formula yields: \(\sqrt{{(\frac{1}{3}-\frac{7}{3})}^2+{(\frac{6}{5}-\frac{1}{5})}^2}\)

Step 3: Simplify the expression

Simplify each subtraction inside the square roots: \(\sqrt{{(-2)}^2+1^2} = \sqrt{4+1} = \sqrt{5}\).

Step 4: Convert the radical to decimal form

\(\sqrt{5}\) is approximately \(2.23607\). Round this value to two decimal places, yielding \(2.24\).