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Chapter 2: Problem 4

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimals places. $$(2,-3) \text { and }(-1,5)$$

Short Answer

Expert verified
The distance between the points (2,-3) and (-1,5) is approximately \(8.54\).

Step by step solution

01

Analyzing the exercise

The problem is about finding the distance between two points in a 2D space. The two points given are (2,-3) and (-1,5). The distances on the x and y axes are both crucial as they form the sides of a right triangle, with the distance between the points being the hypotenuse.
02

Apply the distance formula

The distance 'd' between two points (x1,y1) and (x2, y2) in a 2D space can be calculated using the following formula: \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\). Substitute the given points into the formula. So, here \(x1 = 2\), \(y1 = -3\), \(x2 = -1\), and \(y2 = 5\). Hence, the distance \(d = \sqrt{(-1-2)^2 + (5-(-3))^2}\).
03

Solve the expression

Compute the expression under the square root first. This simplifies to \(d = \sqrt{(-3)^2 + (8)^2} = \sqrt{9+64} = \sqrt{73}\).
04

Convert the simplified radical to decimal form

The simplified radical form \(\sqrt{73}\) can be converted to decimal form using a calculator. Hence, \(d \approx 8.54\), rounded to two decimal places.

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Most popular questions from this chapter

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Exercises \(103-105\) will help you prepare for the material covered in the next section. Solve by completing the square: \(y^{2}-6 y-4=0\).
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