# Analytical Geometry

• April 2nd 2008, 08:01 PM
Iux213
Analytical Geometry
I am confused with the following question... 3. Find the shortest distance from the given point to the given line.

b) (5,0) and y = 0.5x+5. I know that you have to find the perpendicular slope..and then write out the equation for that, I'm just not sure how to solve the question, or for that matter, do the entire question...
• April 2nd 2008, 08:13 PM
math_science_dude
Iux,

I will try to help you out. The distance between the given point and the line will be determined by the length of the line sement connecting that POINT to the LINE (in your question), and perpendicular to the LINE. To that end, you want to find the equation of a line perpendicular to the LINE and which passes through the POINT. Do you recall the relationship between the slope of lines that are perpendicular to one another (i.e. m1*m2 = -1). If you have the slope and the point the line passes through you can determine the equation of the line. Then just find the intersecting point of this line with your LINE. Use the distance formula between this point and your POINT. wallah!

m_s_d
• April 2nd 2008, 08:34 PM
Gusbob
There is actually a formula for this.

The perpendicular distance from a point to a line is the shortest distance between them. The formula is this:

perpendicular distance $= \frac{|ax_1 + by_1 + c|}{\sqrt {a^2+b^2}}$

where the line is $ax + by + c = 0$ and the point is $(x_1,y_1)$

The proof of this formula is quite long, but if you want to see it: Perpendicular Distance from a Point to a Line