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Mathematics LibreTexts

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


Example \(\PageIndex{1}\)

Find the distance between the points \((1,1)\) and \((-4,3)\).


\[\begin{align*} \text{Distance} &=\sqrt{(-4-1)^2+(3-1)^2} \\[5pt] &=\sqrt{25+4}\\ [5pt] &=\sqrt{29}. \end{align*}\]

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?


\[\begin{align*} M &= \left(\dfrac{3+18}{2}, \dfrac{4+22}{2}\right) \\[5pt] &=(10.5,13). \end{align*}\]


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

  2. 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:

  1. \(y = 2x - 3\) (Use graph then y(x) =)

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

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

  4. \(y = 2x + \{-1,0,1,2,3,5\}\) (find the curly braces "{" and "}" use the list feature)