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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

\[\text{Distance}=\sqrt{(c-a)^2+(d-b)^2}.\]

Example 1

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

Solution

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

The Midpoint Formula

Definition: Midpoint

For points \((a,b)\) and \((c,d)\) the midpoint of the line segment formed by these points has coordinates:

\[M=\Big(\dfrac{a+c}{2},\dfrac{b+d}{2}\Big). \]

Example 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:

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

Exercise

  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)

Contributors

  • Integrated by Justin Marshall.