1.5: The Plane

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Oct 6, 2021
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- 1.4: Composition and Inverses
- 1.6: Applications

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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

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.

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The Distance Formula

The Midpoint Formula

Graphing on a Calculator

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