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) |

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?

Re: Finding point on a line closest to another line

So a line orthogonal to x + y - 2 = 0?

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?

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?

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.

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.

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.