Show that if $M$ is the midpoint of the line segment with endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right),$ then $d(P, M)+d(M, Q)=d(P, Q)$ and $d(P, M)=d(M, Q)$
Added by Angela H.
Step 1
Step 1: Given that $M$ is the midpoint of the line segment with endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$, we can find the coordinates of $M$ as $M\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$. Show more…
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Show that if M is the midpoint of the line segment with endpoints P(x1,y1) and Q(x2,y2), then d(P,M) + d(M,Q) = d(P,Q) and d(P,M) = d(M,Q). To prove d(P,M) + d(M,Q) = d(P,Q), first find the d(P,Q). Suppose that P(x1,y1) and Q(x2,y2) are two points in a coordinate plane, determine d(P,Q) using the distance formula. Choose the correct answer below. A. d(P,Q) = √((x1 - y1)² + (x2 - y2)²) B. d(P,Q) = √((x2 - x1)² + (y2 - y1)²) C. d(P,Q) = √((x1 - y2)² + (x2 - y1)²)
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$M$ is the midpoint of line segment $P Q,$ and $R$ is the midpoint of $T S .$ If $\overrightarrow{P M}=\overrightarrow{M Q}=\vec{a}$ and $\overrightarrow{T R}=\overrightarrow{R S}=\vec{b},$ as shown, prove that $2 \overrightarrow{R M}=\overrightarrow{T P}+\overrightarrow{S Q}$
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Prove the midpoint formula by showing that i) the distance from $\left(x_{1}, y_{1}\right)$ to $$ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ equals the distance from $\left(x_{2}, y_{2}\right)$ to $$ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ and ii) the points $$ \left(x_{1}, y_{1}\right),\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ and $$ \left(x_{2}, y_{2}\right) $$ lie on the same line (see Exercise $86)$
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