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Chapter 2: Problem 5

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((-1,2),(5,4)\)

Short Answer

Expert verified
The points are plotted on a coordinate plane, the distance between the points (-1,2) and (5,4) is \(2\sqrt{10}\), and the midpoint of the line segment joining them is (2,3).

Step by step solution

01

Plot the Points

To plot the points (-1,2) and (5,4), draw a coordinate plane and mark these points. The first number of the pair is the x-coordinate, and the second number is the y-coordinate. So, for (-1,2), move 1 unit left (negative direction on x-axis) and 2 units up. For (5,4), move 5 units right and 4 units up.
02

Distance Between the Points

Now that the points are plotted, let's calculate the distance between them using the distance formula in a two-dimensional space: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Substitute the values, \(x_1 = -1, y_1 = 2, x_2 = 5, y_2 = 4\), into the equation. So the distance \(d = \sqrt{(5-(-1))^2 + (4-2)^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{40} = 2\sqrt{10}\).
03

Line Segment Midpoint

Lastly, to find the midpoint of the line segment, use the midpoint formula for two dimensions, \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). Substitute the values you have, which results in \(M = \left(\frac{-1+5}{2}, \frac{2+4}{2}\right) = (2,3)\). The point (2,3) is the midpoint of the line segment between the points (-1,2) and (5,4).

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