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# Distance Formula in the Coordinate Plane

## Length between two points using a right triangle.

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Distance Formula in the Coordinate Plane

### The Distance Formula

The shortest distance between two points is a straight line. This distance can be calculated by using the distance formula. The distance between two points (x1, y1)\begin{align*}(x_1, \ y_1)\end{align*} and (x2, y2)\begin{align*}(x_2, \ y_2)\end{align*} can be defined as d=(x2x1)2+(y2y1)2\begin{align*}d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\end{align*}. Let’s extend this concept to the shortest distance between a point and a line.

Just by looking at a few line segments from A\begin{align*}A\end{align*} to line l\begin{align*}l\end{align*}, we can tell that the shortest distance between a point and a line is the perpendicular line between them. Therefore, AD\begin{align*}AD\end{align*} is the shortest distance between A\begin{align*}A\end{align*} and line l\begin{align*}l\end{align*}.

#### Calculating the Distance between Two Points

Find the distance between (4, -2) and (-10, 3).

Plug in (4, -2) for (x1, y1)\begin{align*}(x_1, \ y_1)\end{align*} and (-10, 3) for (x2, y2)\begin{align*}(x_2, \ y_2)\end{align*} and simplify.

d=(104)2+(3+2)2=(14)2+(52)Distances are always positive!=196+25=22114.87 units\begin{align*}d& = \sqrt{(-10-4)^2+(3+2)^2}\\ & = \sqrt{(-14)^2+(5^2)} \qquad \qquad \quad Distances \ are \ always \ positive!\\ & = \sqrt{196+25}\\ & = \sqrt{221} \approx 14.87 \ units\end{align*}

#### Finding a Point given the Distance from Another

The distance between two points is 4 units. One point is (1, -6). What is the second point? You may assume that the second point is made up of integers.

We will still use the distance formula for this problem, however, we know d\begin{align*}d\end{align*} and need to solve for (x2, y2)\begin{align*}(x_2, \ y_2)\end{align*}.

416=(1x2)2+(6y2)2=(1x2)2+(6y2)2\begin{align*}4 & = \sqrt{\left (1-x_2 \right )^2+(-6-y_2)^2}\\ 16 & = (1-x_2)^2 + (-6-y_2)^2\end{align*}

At this point, we need to figure out two square numbers that add up to 16. The only two square numbers that add up to 16 are 16+0\begin{align*}16 + 0\end{align*}.

1x2-x2x216=(1x2)242+(-6y2)202=±4=-5 or 3=5 or -3and-6y2=0-y2=6y2=-6\begin{align*}& 16 = \underbrace{ (1-x_2)^2 }_{4^2} + \underbrace{ (\text{-}6-y_2)^2 }_{0^2}\\ 1-x_2 &= \pm 4 \quad && \! \! \! \! \! \! \! \! \! \! \! \text{-}6-y_2=0\\ \text{-}x_2 &= \text{-}5 \ \text{or} \ 3 \quad && \! \! \text{-}y_2=6\\ x_2 &= 5 \ \text{or} \ \text{-}3 \quad \qquad \text{and} \qquad && y_2=\text{-}6\end{align*}

\begin{align*}\ \\ \text{OR} \\ \ \\ 16 & = \underbrace{ (1-x_2)^2 }_{0^2} + \underbrace{ (\text{-}6-y_2)^2 }_{4^2}\\ 1-x_2 & =0 \qquad \quad \qquad \! \text{-}6-y_2=\pm 4\\ \text{-}x_2 & =\text{-}1 \qquad \qquad \qquad \text{-}y_2=10 \ \text{or} \ 2\\ x_2 & =1 \qquad \text{and} \qquad \qquad \! \! \! \! y_2= \text{-}10 \ \text{or} \ \text{-}2\end{align*}

Therefore, the second point could have 4 possibilities: (5, -6), (-3, -6), (1, -10), and (1, -2).

#### Determining the Shortest Distance Between a Point and a Line

Determine the shortest distance between the point (1, 5) and the line \begin{align*}y=\frac{1}{3}x-2\end{align*}.

First, graph the line and point. Second determine the equation of the perpendicular line. The opposite sign and reciprocal of \begin{align*}\frac{1}{3}\end{align*} is -3, so that is the slope. We know the line must go through the given point, (1, 5), so use that to find the \begin{align*}y-\end{align*}intercept.

\begin{align*}y&=-3x+b\\ 5&=-3(1)+b \qquad \quad \text{The equation of the line is} \ y=-3x+8.\\ 8&=b\end{align*}

Next, we need to find the point of intersection of these two lines. By graphing them on the same axes, we can see that the point of intersection is (3, -1), the green point.

Finally, plug (1, 5) and (3,-1) into the distance formula to find the shortest distance.

\begin{align*}d& =\sqrt{(3-1)^2+(-1-5)^2}\\ &=\sqrt{(2)^2+(-6)^2}\\ & = \sqrt{2+36}\\ & = \sqrt{38} \approx 6.16 \ units\end{align*}

### Examples

#### Example 1

Find the distance between (-2, -3) and (3, 9).

Use the distance formula, plug in the points, and simplify.

\begin{align*}d & = \sqrt{(3-(-2))^2 + (9-(-3))^2}\\ & = \sqrt{(5)^2 + (12)^2}\\ & = \sqrt{25+144}\\ & = \sqrt{169} = 13 \ units\end{align*}

#### Example 2

Find the distance between (12, 26) and (8, 7).

Use the distance formula, plug in the points, and simplify.

\begin{align*}d & = \sqrt{(8-12)^2 + (7-26)^2}\\ & = \sqrt{(-4)^2 + (-19)^2}\\ & = \sqrt{16+361}\\ & = \sqrt{377} \approx 19.42 \ units\end{align*}

#### Example 3

Find the shortest distance between (2, -5) and \begin{align*}y=-\frac{1}{2}x+1\end{align*}

Find the slope of the perpendicular line. The opposite reciprocal of \begin{align*}-\frac{1}{2}\end{align*} is \begin{align*}2\end{align*}. We know the perpendicular line has a slope of \begin{align*}2\end{align*} and contains the point (2, -5), so \begin{align*}-5=2(2)+b\end{align*} and therefore \begin{align*}b=-9\end{align*}. Next, we need to figure out where the lines \begin{align*}y=-\frac{1}{2}x+1\end{align*} and \begin{align*}y=2x-9\end{align*} intersect:

\begin{align*}2x-9&=-\frac{1}{2}x+1\\ 2.5x&=10\\ x&=4, \text{and therefore} \ y=-1\end{align*}

So the lines intersect at the point (4, -1). Now, use the distance formula to find the distance between (4, -1) and (2, -5): \begin{align*}\sqrt{(2-4)^2+(-5-(-1))^2}=\sqrt{4+36}=\sqrt{40}\approx 6.32 \text{ units.}\end{align*}

### Review

Find the distance between each pair of points. Round your answer to the nearest hundredth.

1. (4, 15) and (-2, -1)
2. (-6, 1) and (9, -11)
3. (0, 12) and (-3, 8)
4. (-8, 19) and (3, 5)
5. (3, -25) and (-10, -7)
6. (-1, 2) and (8, -9)
7. (5, -2) and (1, 3)
8. (-30, 6) and (-23, 0)

Determine the shortest distance between the given line and point. Round your answers to the nearest hundredth.

1. \begin{align*}y=\frac{1}{3}x+4; \ (5, \ -1)\end{align*}
2. \begin{align*}y = 2x-4; \ (-7, \ -3)\end{align*}
3. \begin{align*}y=-4x+1; \ (4, \ 2)\end{align*}
4. \begin{align*}y=-\frac{2}{3}x-8; \ (7, \ 9)\end{align*}
5. The distance between two points is 5 units. One point is (2, 2). How many possibilities are there for the second point? You may assume that the second point is made up of integers.
6. What if in the previous question the second point was not necessarily made up of integers? What shape would be created by plotting all of the possibilities for the second point?
7. Find one possibility for the equation of a line that is exactly 6 units away from the point (-3,2).

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### Vocabulary Language: English

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be defined as $d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.

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