Given one endpoint and midpoint in (x,y) of a line segment, explain how to find the other end point.

Question

A line segment with one end at C(6,5)has midpoint M(4,2). Determine the coordinates of the other endpoint, D. Explain your solution and describe a method to check your answer.

Answer 1

It's very very simple:

The midpoint formula give you: $$ M=(x_m,y_m)=\left( \dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right) $$ where $(x_1,y_1)$ are the coordinates of a point, say your $C$, and $(x_2,y_2)$ are the coordinate of the oter point, i.e the point $D$ that you are searching.

So, substituting the given coordinates you have:

$$ 4=\dfrac{6+x_2}{2} \quad \land \quad 2=\dfrac{5+y_2}{2} $$ Now can you solve these two simple equations?

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$$ 4\cdot 2=6+x_2 \Rightarrow 8-6=x_2 \Rightarrow x_2=2 $$ $$ 2 \cdot 2=5+y_2 \Rightarrow 4-5=y_2 \Rightarrow y_2=-1 $$