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Chapter 1: Problem 21

Find the distance between the points. $$\left(\frac{1}{2}, \frac{4}{3}\right),(2,-1)$$

Short Answer

Expert verified
The distance between the two points is \(\frac{\sqrt{277}}{6}\).

Step by step solution

01

Identify the coordinates of the points

The given points are \((\frac{1}{2}, \frac{4}{3})\) and \((2,-1)\). Identify and label these coordinates. The first point can be labeled as \(P_1(\frac{1}{2}, \frac{4}{3})\) and the second point as \(P_2(2, -1)\). Accordingly, \(x_1 = \frac{1}{2}\), \(y_1 = \frac{4}{3}\), \(x_2 = 2\), and \(y_2 = -1\).
02

Apply the Distance Formula

The Distance Formula is \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]. Substituting the identified coordinates into the distance formula gives \[d = \sqrt{(2-\frac{1}{2})^2 + (-1-\frac{4}{3})^2}\]
03

Simplify the Expression to find Distance

Simplify the expression to find the distance. That gives \[d = \sqrt{(\frac{3}{2})^2 + (\frac{-7}{3})^2} = \sqrt{\frac{9}{4} + \frac{49}{9}}\]. To simplify further, find a common denominator and add the fractions, then take the square root of the result. That gives \[d = \sqrt{\frac{81}{36} + \frac{196}{36}} = \sqrt{\frac{277}{36}} = \frac{\sqrt{277}}{6}\]

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