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Chapter 0: Problem 66

Find the distance between the points. a. \((1,3)\) and \((4,7)\) b. \((1,0)\) and \((4,4)\) c. \((-1,3)\) and \((4,9)\) d. \((-2,1)\) and \((10,6)\)

Short Answer

Expert verified
The distances between the given points are as follows: a. \(5\) units b. \(5\) units c. \(\sqrt{61}\) units d. \(13\) units

Step by step solution

01

(a. Distance between \((1,3)\) and \((4,7)\))

Let's denote point A as \((1,3)\) and point B as \((4,7)\). To find the distance between them, we will apply the distance formula: \(d = \sqrt{(4 - 1)^2 + (7 - 3)^2}\) \(d = \sqrt{3^2 + 4^2}\) \(d = \sqrt{9 + 16}\) \(d = \sqrt{25}\) \(d = 5\) The distance between points A and B is 5 units.
02

(b. Distance between \((1,0)\) and \((4,4)\))

Let's denote point A as \((1,0)\) and point B as \((4,4)\). To find the distance between them, we will apply the distance formula: \(d = \sqrt{(4 - 1)^2 + (4 - 0)^2}\) \(d = \sqrt{3^2 + 4^2}\) \(d = \sqrt{9 + 16}\) \(d = \sqrt{25}\) \(d = 5\) The distance between points A and B is 5 units.
03

(c. Distance between \((-1,3)\) and \((4,9)\))

Let's denote point A as \((-1,3)\) and point B as \((4,9)\). To find the distance between them, we will apply the distance formula: \(d = \sqrt{(4 - (-1))^2 + (9 - 3)^2}\) \(d = \sqrt{5^2 + 6^2}\) \(d = \sqrt{25 + 36}\) \(d = \sqrt{61}\) The distance between points A and B is \(\sqrt{61}\) units.
04

(d. Distance between \((-2,1)\) and \((10,6)\))

Let's denote point A as \((-2,1)\) and point B as \((10,6)\). To find the distance between them, we will apply the distance formula: \(d = \sqrt{(10 - (-2))^2 + (6 - 1)^2}\) \(d = \sqrt{12^2 + 5^2}\) \(d = \sqrt{144 + 25}\) \(d = \sqrt{169}\) \(d = 13\) The distance between points A and B is 13 units.

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