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Chapter 2: Problem 14

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimals places. $$(0,-\sqrt{2}) \text { and }(\sqrt{7}, 0)$$

Short Answer

Expert verified
The distance between the points ((0, -\(\sqrt{2}\)) and (\(\sqrt{7}\), 0)) is 3.00

Step by step solution

01

Identify and label the points

The given points are (0, -\(\sqrt{2}\)) and (\(\sqrt{7}\), 0). We'll label them as follows: Point A = (0, -\(\sqrt{2}\)) = (x1, y1) and Point B = (\(\sqrt{7}\), 0) = (x2, y2).
02

Apply the distance formula

Now, calculate the distance \(d\) between the points A and B using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting the values, we get \(d = \sqrt{\{(sqrt{7} - 0)^2 + (0 - (-sqrt{2}))^2}\}\).
03

Simplify the radical form

Simplify the expression under the square root to obtain the distance in simplified radical form. The squared terms will give us 7 and 2, so the equation becomes \(d = sqrt{7 + 2}\). Thus, the simplified radical form of the distance is \(d = sqrt{9}\).
04

Calculate the numerical distance

Finally, we calculate the square root of 9 which gives us 3. This is the exact distance between points A and B.
05

Round to two decimal places

As the distance between the points is an integer, rounding it to two decimal places will still give 3.00.

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