A site needs to be located the same distance from the two main towns that have coordinates (3,10) and (13,4). Determine an equation for the line that shows all the possible sites for the fair.

Question

A committee is choosing a site for a country fair. The site needs to be located the same distance from the two main towns in the country. On a map, these towns have coordinates (3,10) and (13,4). Determine an equation for the line that shows all the possible sites for the fair.

Answer 1

If the required point is joined to the two locations, an isosceles triangle is formed. The perpendicular from the required point on to the line joining the two locations bisects this line and is perpendicular to it. So the answer is to find the equation of the perpendicular which passes through the midpoint.

The slope of the line between the two locations is: (4-10)/(13-3)=-6/10=-⅗. The slope of the perpendicular is 5/3.

The midpoint is ((3+13)/2,(10+4)/2)=(8,7).

Therefore y-7=5(x-8)/3 is the equation of the line, y=5x/3-40/3+7, y=(5x-19)/3.