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1.5: The Plane

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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 $$\PageIndex{1}$$

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

Solution

\begin{align*} \text{Distance} &=\sqrt{(-4-1)^2+(3-1)^2} \\[4pt] &=\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) \\[4pt] &=(10.5,13). \end{align*}

Exercises

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)