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Chapter 2: Problem 29

Find the midpoint of each line segment with the given endpoints. $$(\sqrt{18},-4) \text { and }(\sqrt{2}, 4)$$

Short Answer

Expert verified
The midpoint of the line segment with endpoints \((\sqrt{18},-4)\) and \((\sqrt{2},4)\) is \((2\sqrt{2}, 0)\).

Step by step solution

01

Identify the endpoints

The problem provides us with the endpoints of the line segment, which are \((\sqrt{18},-4)\) and \((\sqrt{2},4)\). Hence, let's identify \(x_1 = \sqrt{18}\), \(y_1 = -4\), \(x_2 = \sqrt{2}\), and \(y_2 = 4\).
02

Apply the Midpoint Formula

The formula for the midpoint M, given two points \((x_1, y_1)\) and \((x_2, y_2)\), is given by \(M = \left( \frac{x_1+ x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Substituting the values we identified in the previous step, we get \(M = \left( \frac{\sqrt{18}+ \sqrt{2}}{2}, \frac{-4 + 4}{2} \right)\).
03

Simplify the results

Let's simplify the coordinate values of the midpoint. Simplification yields \(M = \left( \frac{\sqrt{18}+ \sqrt{2}}{2}, 0 \right)\). The factorial simplification of \(\sqrt{18}\) is \(3\sqrt{2}\) which makes the calculation easier. So the simplified midpoint becomes \(M = \left( 2\sqrt{2}, 0 \right)\).

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