MHB How to Find the Perpendicular Distance from a Point to a Line?

[mathdad]

Use the distance formula to find the perpendicular distance from P(1, 3) to the line y = (x/2) - 5.

Any ideas on how to get started?

 

[topsquark]

Use the distance formula to find the perpendicular distance from P(1, 3) to the line y = (x/2) - 5.

Any ideas on how to get started?

Try starting with this.

-Dan

 

[Prove It]

Use the distance formula to find the perpendicular distance from P(1, 3) to the line y = (x/2) - 5.

Any ideas on how to get started?

Perpendicular lines have gradients that multiply to give -1, so the gradient of the line you are looking for is -2. You know that (1, 3) lies on this line, so

$\displaystyle \begin{align*} y - 3 &= -2 \left( x - 1 \right) \\ y - 3 &= -2\,x + 2\\ y &= -2\,x + 5 \end{align*}$

And now you want to know where $\displaystyle \begin{align*} y = -2\,x + 5 \end{align*}$ and $\displaystyle \begin{align*} y = \frac{x}{2} - 5 \end{align*}$ intersect, so that you can work out the distance between this point and P(1, 3).

 

[mathdad]

Perpendicular lines have gradients that multiply to give -1, so the gradient of the line you are looking for is -2. You know that (1, 3) lies on this line, so

$\displaystyle \begin{align*} y - 3 &= -2 \left( x - 1 \right) \\ y - 3 &= -2\,x + 2\\ y &= -2\,x + 5 \end{align*}$

And now you want to know where $\displaystyle \begin{align*} y = -2\,x + 5 \end{align*}$ and $\displaystyle \begin{align*} y = \frac{x}{2} - 5 \end{align*}$ intersect, so that you can work out the distance between this point and P(1, 3).

To find where the two lines meet, do I set the equations equal to each other?

 

[MarkFL]

To find where the two lines meet, do I set the equations equal to each other?

Yes, set:

$$-2x+5=\frac{x}{2}-5$$

and solve for $x$ to get the $x$-coordinate of the intersection point. Then plug this value for $x$ into either line (doesn't matter which as they will give the same $y$ value) to get the $y$-coordinate. :D

 

[mathdad]

Yes, set:

$$-2x+5=\frac{x}{2}-5$$

and solve for $x$ to get the $x$-coordinate of the intersection point. Then plug this value for $x$ into either line (doesn't matter which as they will give the same $y$ value) to get the $y$-coordinate. :D

I can do it. Thanks.