Question
Midpoint of a line segment Show that the point with coordinates$$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$$is the midpoint of the line segment joining $P\left(x_{1}, y_{1}\right)$ to $Q\left(x_{2}, y_{2}\right)$
Step 1
Step 1: First, we are given two points P and Q with coordinates $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$ respectively. Show more…
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Midpoint of a line segment Use vectors to show that the midpoint of the line segment joining $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is the point $\left(\left(x_{1}+x_{2}\right) / 2,\left(y_{1}+y_{2}\right) / 2\right) .$ (Hint: Let $O$ be the origin and let $M$ be the midpoint of $P Q .$ Draw a picture and show that $\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P}) \cdot)$
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Midpoint formula Prove that the midpoint of the line segment joining $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ is $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$
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a. Show that the midpoint of the line segment joining the points $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right)$ is $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$$ b. Use the result of part (a) to find the midpoint of the line segment joining the points (-3,2) and (4,-5).
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