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Chapter 1: Problem 31

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. $$(1,1),(9,7)$$

Short Answer

Expert verified
The points plotted on a 2D Cartesian coordinate system will yield a line segment. The distance between the points (1,1) and (9,7) is 10 units. The midpoint of the line segment joining these points is at the point (5,4).

Step by step solution

01

Plotting the Points

Plot the points given on a 2D Cartesian coordinate system. The first value in each pair represents the x-coordinate and the second value represents the y-coordinate. For instance, for point (1,1), move 1 unit to the right along the x-axis from the origin and then move 1 unit up from there to get the point (1,1). Repeat this process for point (9,7). Join these points with a line segment to visualize clearly.
02

Calculating the Distance

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a 2D Cartesian coordinate system is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). In this case, \((x_1, y_1) = (1,1)\) and \((x_2, y_2) = (9,7)\). Substituting these values into the formula gives the distance as \(\sqrt{(9 - 1)^2 + (7 - 1)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\) units
03

Finding the Midpoint

The midpoint of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\). Here, \((x_1, y_1) = (1,1)\) and \((x_2, y_2) = (9,7)\). Substituting these values into the formula gives the midpoint as \((\frac{1 + 9}{2}, \frac{1 + 7}{2}) = (5, 4)\)

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