# Finding point on a line closest to another line

• Jun 13th 2013, 12:46 PM
biggerleaffan
Finding point on a line closest to another line
 Find the point on the line x + y − 2 = 0 that is closest to the point (-2, 5)
• Jun 13th 2013, 12:49 PM
MarkFL
Re: Finding point on a line closest to another line
What you want is the intersection of the line perpendicular to the given line which passes through the given point. Can you proceed?
• Jun 13th 2013, 01:02 PM
biggerleaffan
Re: Finding point on a line closest to another line
So a line orthogonal to x + y - 2 = 0?
• Jun 13th 2013, 01:10 PM
Plato
Re: Finding point on a line closest to another line
Quote:

Originally Posted by biggerleaffan
So a line orthogonal to x + y - 2 = 0?

Note that \$\displaystyle x-y+7=0\$ is a line through \$\displaystyle (-2,5)\$ and perpendicular to the given line.

Where do they intersect?
• Jun 13th 2013, 01:32 PM
biggerleaffan
Re: Finding point on a line closest to another line
Quote:

Originally Posted by Plato
Note that \$\displaystyle x-y+7=0\$ is a line through \$\displaystyle (-2,5)\$ and perpendicular to the given line.

Where do they intersect?

-5/2, 9/2.

Is this then the answer to my original question?
• Jun 13th 2013, 01:54 PM
Plato
Re: Finding point on a line closest to another line
Quote:

Originally Posted by biggerleaffan
-5/2, 9/2.
Is this then the answer to my original question?

I don't know if that is correct. I will not do the algebra.

But by definition: the distance from a point to a line is the length of the line segment from the point to the line.
• Jun 13th 2013, 03:57 PM
HallsofIvy
Re: Finding point on a line closest to another line
Quote:

Originally Posted by Plato
I don't know if that is correct. I will not do the algebra.

But by definition: the distance from a point to a line is the length of the line segment from the point to the line.

There are many "line segments from the point to the line".

By definition: the distance from a point to a line is the length of the line segment, perpendicular to given line, from the point to the line.
• Jun 13th 2013, 04:56 PM
Plato
Re: Finding point on a line closest to another line
Quote:

Originally Posted by HallsofIvy
There are many "line segments from the point to the line".

By definition: the distance from a point to a line is the length of the line segment, perpendicular to given line, from the point to the line.

RIGHT! I have no idea where my "perpendicular" went to. Clearly a typo. Thank you.