Question 1185691: Given that point A is (-2,4), C (4,7), and B (2,-2), what is the shortest distance from B to AC?
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39625)   (Show Source):
You can put this solution on YOUR website! Line AC is  .
Line through B(2,-2) and perpendicular to line AC is,....  .
Intersection point of these two lines on line AC is,..... (-6/5, 22/5).
Shortest distance from B to line AC is the distance between B(2,-2) and (-6/5, 22/5). You can use the Distance Formula.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
The solution method shown by the other tutor is fine.
Here is an alternative method. If you are solving this kind of problem (finding the shortest distance from a point to a line) frequently, this method will be faster.
(1) Find the equation of line AC in the form Ax+By+C=0.
From A to C the change in x is +6 and the change in y is +3, so the slope is 3/6 = 1/2. So the equation is of the form y=(1/2)x+b.
Plug in either of the given (x,y) values to determine b.
4=(1/2)(-2)+b
4=-1+b
b=5
The equation of the line in slope-intercept form is y=(1/2)x+5.
Put the equation in the required form.
y=(1/2)x+5
2y=x+10
x-2y+10=0
(2) Use the formula for the (shortest) distance from a point (p,q) to the line with equation Ax+By+C=0:
ANSWER: 7.1554 to 4 decimal places
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