SOLUTION: determine the ratio in which the line joining (0,7) and (-2,1)is divided by the line 2x+y-4=0

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Question 65750: determine the ratio in which the line joining (0,7) and (-2,1)is
divided by the line 2x+y-4=0
Found 2 solutions by Nate, Edwin McCravy:
Answer by Nate(3500)   (Show Source): You can put this solution on YOUR website!
Ratio of the slopes?
m = (7 - 1)/(0 + 2) = 6/2 = 3
y - 7 = 3(x + 0)
y = 3x + 7 is the equation just in case ....
second:
y = -2x + 4
slope: -2
Ratio of slopes: 3:-2 or 3/-2
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
determine the ratio in which the line joining (0,7) and (-2,1)is 
divided by the line 2x+y-4=0

Nate above took it that you wanted the ratio of the slopes. I think
you wanted the ratio in which the line divides the segment.


Here's a graph of the line segment joining (0,7) and (-2,1) in red



Now we'll add the graph of 2x + y - 4 = 0 in green



We want to know into what ratio the green line divides the red 
line segment.

Plan:

1. Find the equation of the line joining (0,7) and (-2,1)
2. Solve the system of equations consisting of the results 
   of step 1 and line 2x+y-4=0 to find their point of 
   intersection.
3. Find the distances from the point found in step 3 to 
   (0,7) and (-2,1) 
4. Find the ratio of these two distances.

1. Find the slope, m:

     y2 - y1
m = —————————
     x2 - x1

where (x1, y1) = (0,7) and (x2, y2) = (-2,1)

     (1) - (7)     -6     
m = ——————————— = ———— = 3
     (-2) - (0)    -2      

Now substitute in the point slope formula:

y - y1 = m(x - x1)

y - 7 = 3(x - 0)
y - 7 = 3x
    y = 3x + 7

2. Solve the system:
  
    2x + y - 4 = 0
    y = 3x + 7

You can do this by substitution.  
Point of intersection = (-3/5, 26/5) = (-.6, 5.2)

3. Use the distance formula
        _____________________
   d = Ö(x2 - x1)2+(y2 - y1)2

 to find the distance from (0,7) to (-.6, 5.2)

        _____________________
   d = Ö(-.6 - 0)2+(5.2 - 7)2
        __________
   d = Ö.36 + 3.24
        ___
   d = Ö3.6

Also use it to find the distance from (-.6, 5.2) to (-2,1)
        _________________________
   d = Ö(-2 - (-.6) )2+(1 - 5.2)2
        ____________________
   d = Ö(-2 + .6)2 + (-4.2)2
        _______________
   d = Ö(-1.4)2 + 17.64
        ____________ 
   d = Ö1.96 + 17.64
        ____
   d = Ö19.6
                                    
4. The ratio of the two distances "longer to shorter" is
    ____     ___
i  Ö19.6 to Ö3.6, which can be
   expressed as a fraction
     ________     ______    ___  __
    Ö19.6/3.6  = Ö196/36 = Ö196/Ö36 = 14/6 = 7/3 or 7 to 3 or 7:3.

The ratio "shorter to longer" is 3 to 7 or 3:7

Edwin
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