 3.05 Extension: Perpendiculars and distance

Lesson

In Algebra 1, you should have become familiar with the standard form of a linear equation: $Ax+By=C$Ax+By=C.

Another form for the equation of a line that is very similar to standard form is called the general form. Essentially it is the same as the standard form, except the constant $C$C now is found on the other side of the equation, and one side has been set equal to zero. General form looks like:

$Ax+By+C=0$Ax+By+C=0

All coefficients $A$A,$B$B and $C$C are integers and the coefficient of $x$x is positive.

We've also studied how to find the distance between two points on a number plane, using the Pythagorean Theorem or the distance formula: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$d=(x2x1)2+(y2y1)2.

Perpendicular distance

The perpendicular distance is the shortest distance between a point and a line. The formula for perpendicular distance can be derived from the two formulas reviewed above: General form for the equation of a line, and the distance formula.

Exploration

Let's start with the line $Ax+By+C=0$Ax+By+C=0 and label it $\overleftrightarrow{DE}$DE. It has slope of $\frac{-A}{B}$AB. Now, let's draw on a point $P$P with coordinates $\left(m,n\right)$(m,n). We want to find the perpendicular distance from the point $P$P to $\overleftrightarrow{DE}$DE (that is, the length $PQ$PQ). Now, we're going to draw a second line that is parallel to $\overleftrightarrow{DE}$DE (so it has the same slope) and passes through point $P$P. This line has been labeled $\overleftrightarrow{FG}$FG. Now we will construct another line segment parallel to $\overline{PQ}$PQ that passes through the origin. This line will have slope $\frac{B}{A}$BA because it is perpendicular to $\overleftrightarrow{DE}$DE. Let's call this new length $RS$RS

We will find the length $RS$RS, which will be the same length $PQ$PQ that we wanted at the start. Ok, now we've done the prep work, let's do some calculations!

Using the point-slope formula for the equation of a line and rearranging it, we can write the equation of $\overleftrightarrow{FG}$FG as $y=\frac{-Ax+Am+Bn}{B}$y=Ax+Am+BnB.

Similarly, line $\overleftrightarrow{RS}$RS has the equation $y=\frac{B}{A}x$y=BAx.

Now let's find the point of intersection between $\overleftrightarrow{FG}$FG and $\overleftrightarrow{RS}$RS by using a system of equations.

 $\frac{-Ax+Am+Bn}{B}$−Ax+Am+BnB​ $=$= $\frac{B}{A}x$BA​x (Set the $y$y-values equal to one another) $\frac{A\left(Am+Bn\right)}{A^2+B^2}$A(Am+Bn)A2+B2​ $=$= $x$x (Solve for $x$x) $x$x $=$= $\frac{A\left(Am+Bn\right)}{A^2+B^2}$A(Am+Bn)A2+B2​ $y$y $=$= $\frac{B}{A}\times\frac{A\left(Am+Bn\right)}{A^2+B^2}$BA​×A(Am+Bn)A2+B2​ (Substitute into $y=\frac{B}{A}x$y=BA​x) $=$= $\frac{B\left(Am+Bn\right)}{A^2+B^2}$B(Am+Bn)A2+B2​ (Simplify)

So the coordinates of $R$R are $\left(\frac{A\left(Am+Bn\right)}{A^2+B^2},\frac{B\left(Am+Bn\right)}{A^2+B^2}\right)$(A(Am+Bn)A2+B2,B(Am+Bn)A2+B2).

Using this same process, we can conclude that the coordinates of $S$S are $\left(\frac{-AC}{A^2+B^2},\frac{-BC}{A^2+B^2}\right)$(ACA2+B2,BCA2+B2).

Now let's use the distance formula to find $RS$RS:

 $d$d $=$= $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$√(x2​−x1​)2+(y2​−y1​)2 $=$= $\sqrt{\left(\frac{-AC}{A^2+B^2}-\frac{A\left(Am+Bn\right)}{A^2+B^2}\right)^2+\left(\frac{-BC}{A^2+B^2}-\frac{B\left(Am+Bn\right)}{A^2+B^2}\right)^2}$√(−ACA2+B2​−A(Am+Bn)A2+B2​)2+(−BCA2+B2​−B(Am+Bn)A2+B2​)2 $=$= $\sqrt{\frac{\left(-a\left(Am+Bn+c\right)\right)^2+\left(-B\left(Am+Bn+c\right)\right)^2}{\left(A^2+B^2\right)^2}}$√(−a(Am+Bn+c))2+(−B(Am+Bn+c))2(A2+B2)2​ $=$= $\sqrt{\frac{\left(A^2+B^2\right)\left(Am+Bn+c\right)}{\left(A^2+B^2\right)^2}}$√(A2+B2)(Am+Bn+c)(A2+B2)2​ $=$= $\sqrt{\frac{Am+Bn+c}{A^2+B^2}}$√Am+Bn+cA2+B2​

We can apply the square root but, since distance must be a positive value, will will include the absolute value sign, just in case the numerator is a negative value.

So, the distance from point $\left(x_1,y_1\right)$(x1,y1) to line $Ax+By+C=0$Ax+By+C=0 is given by

$d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2+B^2}}$d=|Ax1+By1+C|A2+B2.

The applet below will help visualize that formula:

The perpendicular distance formula

$d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2+B^2}}$d=|Ax1+By1+C|A2+B2

Practice questions

Question 1

Which of the following formulas is to find the perpendicular distance $d$d from the point $\left(x_1,y_1\right)$(x1,y1) to the line $Ax+By+C=0$Ax+By+C=0?

1. $d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2-B^2}}$d=|Ax1+By1+C|A2B2

A

$d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2+B^2}}$d=|Ax1+By1+C|A2+B2

B

$d=\frac{\left|Ax_1-By_1+C\right|}{\sqrt{A^2+B^2}}$d=|Ax1By1+C|A2+B2

C

$d=\frac{\left|Ax_1-By_1+C\right|}{\sqrt{A^2-B^2}}$d=|Ax1By1+C|A2B2

D

$d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2-B^2}}$d=|Ax1+By1+C|A2B2

A

$d=\frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2+B^2}}$d=|Ax1+By1+C|A2+B2

B

$d=\frac{\left|Ax_1-By_1+C\right|}{\sqrt{A^2+B^2}}$d=|Ax1By1+C|A2+B2

C

$d=\frac{\left|Ax_1-By_1+C\right|}{\sqrt{A^2-B^2}}$d=|Ax1By1+C|A2B2

D

Question 2

Find the perpendicular distance from the origin to the line $4x+7y+5=0$4x+7y+5=0.

Question 3

If the perpendicular distance from the point ($-5$5, $-1$1) to the line $-2x-7y+k=0$2x7y+k=0 is $\frac{10}{\sqrt{53}}$1053, what is the value(s) of $k$k?