Pam is taking train from the town of Rome to the town of Florence. Rome is located 30 miles due West of the town of Paris. Florence is 25 miles East, and 45 mies North of Rome.

Question

On her trip, how close does Pam get to Paris?

My work so far...
She's in Rome due West, so would that make her coordinate point (-30,0)
She's trying to go to Florence which is... (25,45)?
and Paris is (45,0)?
The equation for the line is y=9/11(x+30)
I connect the line to Rome and got a perpendicular line and the equation for that is:
y=-11/9(x+30)
I need help solving it. I don't know what to do next.

Answer 1

I assume you placed Paris at the origin (0,0). Call it P
I agree with the location of R and F
and the equation of RF

Now, the shortest distance line from P to the line RF would have a slope of -11/9, you had that, but it would pass through the origin, so the equation would be
y = -11/9x

so solve (9/11)x + 270/11 and y = (-11/9()x

(9/11)x + 270/11 = (-11/9)x
multiplying by 99, and simplifying I got
x = -1215/101
subbing that back into the simpler of the two equations, I got y = 1485/101

so the closest point is (-1215/101,1485/101)

I then used the distance formula between that point and (0,0) , and my calculator, to find the shortest to be 18.997

A much easier way is to know that if the equation of a line is written as
Ax + By + C = 0, and P(a,b) is any point then the shortest disance from P to the line is
│Aa + Bb + C│/√(a^2 + B^2)

so your line equation is 9x - 11y + 270 = 0
so shortest distance is │0 + 0 + 270│/√(81+121) = 18.997 !!!!!

Every student should know that formula!!