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Chapter 1: Problem 51

(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 distance between the points (-1,2) and (5,4) is \(2\sqrt{10}\) and the midpoint is (2, 3).

Step by step solution

01

Plot the Points

First, plot the points (-1,2) and (5,4) on a set of x-y axes. The point (-1,2) will fall in the second quadrant, while the point (5,4) will fall in the first quadrant.
02

Find Distance between Points

Next, find the distance between the points using the distance formula: \[ d = \sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2 }\]. Here, (-1,2) and (5,4) can be taken as \((x_1,y_1)\) and \((x_2,y_2)\) respectively. So the distance \(d\) is \(\sqrt{ (5-(-1))^2 + (4-2))^2 } = \sqrt{ 36+4 } = \sqrt{ 40 }\), which simplifies to \(2\sqrt{10}\).
03

Find Midpoint between Points

Finally, use the midpoint formula \[\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\] to find the midpoint between (-1,2) and (5,4). This results in \(\left(\frac{-1+5}{2}, \frac{2+4}{2} \right)\), which simplifies to (2, 3).

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