1.5: The Plane
The Distance Formula
Definition: Distance
Recall that for two points \((a,b)\) and \((c,d)\) in a plane, the distance is found by the formula
\[\text{Distance}=\sqrt{(ca)^2+(db)^2}.\]
Example \(\PageIndex{1}\)
Find the distance between the points \((1,1)\) and \((4,3)\).
Solution
\[\begin{align*} \text{Distance} &=\sqrt{(41)^2+(31)^2} \\[5pt] &=\sqrt{25+4}\\ [5pt] &=\sqrt{29}. \end{align*}\]
The Midpoint Formula
Definition: Midpoint
For points \((a,b)\) and \((c,d)\) the midpoint of the line segment formed by these points has coordinates:
\[M=\left(\dfrac{a+c}{2},\dfrac{b+d}{2}\right). \]
Example \(\PageIndex{2}\)
Suppose that you have a boat at one side of the lake with coordinates \((3,4)\) and your friend has a boat at the other side of the lake with coordinates \((18,22)\). If you want to meet half way, at what coordinates should you meet?
Solution:
\[\begin{align*} M &= \left(\dfrac{3+18}{2}, \dfrac{4+22}{2}\right) \\[5pt] &=(10.5,13). \end{align*}\]
Exercises

Show that the points \((5,14)\), \((1,4)\), and \((11,10)\) are vertices of an isosceles triangle.

Show that the triangle with vertices \((1,1)\), \((1,1)\), and \((\sqrt{3},\sqrt{3})\) are vertices of a right triangle.
Graphing on a Calculator
We will graph the equations:

\(y = 2x  3\) (Use graph then y(x) =)

\(y = 5x^2 + 4\)

\(y = x + 1\) (To find absolute value, use catalog then hit enter)

\(y = 2x + \{1,0,1,2,3,5\}\) (find the curly braces "{" and "}" use the list feature)
Contributors
 Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.