SOLUTION: Distance from a Point to a Line 4.Find the distance from A(-2,-2) to the line joining B(5,2) and C(-1,4), to the nearest hundredth. Thanks alot!!!!

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Distance from a Point to a Line 4.Find the distance from A(-2,-2) to the line joining B(5,2) and C(-1,4), to the nearest hundredth. Thanks alot!!!!      Log On


   

Question 189656: Distance from a Point to a Line
4.Find the distance from A(-2,-2) to the line joining B(5,2) and C(-1,4), to the nearest hundredth.
Thanks alot!!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1) Find the equation of the line that passes through the points B(5,2) and C(-1,4)


First let's find the slope of the line through the points and


Note: is the first point and is the second point .


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula.


m=%284-2%29%2F%28-1-5%29 Plug in y%5B2%5D=4, y%5B1%5D=2, x%5B2%5D=-1, and x%5B1%5D=5


m=%282%29%2F%28-1-5%29 Subtract 2 from 4 to get 2


m=%282%29%2F%28-6%29 Subtract 5 from -1 to get -6


m=-1%2F3 Reduce


So the slope of the line that goes through the points and is m=-1%2F3


Now let's use the point slope formula:


y-y%5B1%5D=m%28x-x%5B1%5D%29 Start with the point slope formula


y-2=%28-1%2F3%29%28x-5%29 Plug in m=-1%2F3, x%5B1%5D=5, and y%5B1%5D=2


y-2=%28-1%2F3%29x%2B%28-1%2F3%29%28-5%29 Distribute


y-2=%28-1%2F3%29x%2B5%2F3 Multiply


y=%28-1%2F3%29x%2B5%2F3%2B2 Add 2 to both sides.


y=%28-1%2F3%29x%2B11%2F3 Combine like terms.


y=%28-1%2F3%29x%2B11%2F3 Simplify


So the equation that goes through the points and is y=%28-1%2F3%29x%2B11%2F3


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Step 2) Find the perpendicular slope


Note: if you draw the line and plot the points, you'll find that the shortest distance is along a perpendicular line from the point to the given line.


Take note that the slope of y=%28-1%2F3%29x%2B11%2F3 is m=-1%2F3. Change the sign and flip the fraction to get m=3%2F1 or just m=3


So the perpendicular slope is m=3


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Step 3) Find the equation of the line that has a slope of m=3 (the perpendicular slope) and goes through the given point A(-2,-2)


If you want to find the equation of line with a given a slope of m=3 which goes through the point (-2,-2) you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---

y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is the given point


So lets use the Point-Slope Formula to find the equation of the line


y--2=%283%29%28x--2%29 Plug in m=3, x%5B1%5D=-2, and y%5B1%5D=-2 (these values are given)


y%2B2=%283%29%28x--2%29 Rewrite y--2 as y%2B2


y%2B2=%283%29%28x%2B2%29 Rewrite x--2 as x%2B2


y%2B2=3x%2B%283%29%282%29 Distribute 3


y%2B2=3x%2B6 Multiply 3 and 2 to get 6


y=3x%2B6-2 Subtract 2 from both sides to isolate y


y=3x%2B4 Combine like terms 6 and -2 to get 4



So the equation of the line that has a slope of m=3 goes through (-2,-2) is y=3x%2B4

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Step 4) Find the point of intersection between the two lines y=%28-1%2F3%29x%2B11%2F3 and y=3x%2B4


y=3x%2B4 Start with the second equation.


%28-1%2F3%29x%2B11%2F3=3x%2B4 Plug in y=%28-1%2F3%29x%2B11%2F3


-x%2B11=9x%2B12 Multiply EVERY term by the LCD 3 to clear out the fractions.


-x=9x%2B12-11 Subtract 11 from both sides.


-x-9x=12-11 Subtract 9x from both sides.


-10x=12-11 Combine like terms on the left side.


-10x=1 Combine like terms on the right side.


x=%281%29%2F%28-10%29 Divide both sides by -10 to isolate x.


x=-1%2F10 Reduce.


y=3x%2B4 Go back to the second equation.


y=3%28-1%2F10%29%2B4 Plug in x=-1%2F10


y=-3%2F10%2B4 Multiply


y=37%2F10 Combine like terms.



So the lines intersect at the point


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Step 5) Find the distance between the two points (-2,-2) and



d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula.


d=sqrt%28%28-2--1%2F10%29%5E2%2B%28-2-37%2F10%29%5E2%29 Plug in x%5B1%5D=-2, x%5B2%5D=-1%2F10, y%5B1%5D=-2, and y%5B2%5D=37%2F10.


d=sqrt%28%28-19%2F10%29%5E2%2B%28-2-37%2F10%29%5E2%29 Subtract -1%2F10 from -2 to get -19%2F10.


d=sqrt%28%28-19%2F10%29%5E2%2B%28-57%2F10%29%5E2%29 Subtract 37%2F10 from -2 to get -57%2F10.


d=sqrt%28361%2F100%2B%28-57%2F10%29%5E2%29 Square -19%2F10 to get 361%2F100.


d=sqrt%28361%2F100%2B3249%2F100%29 Square -57%2F10 to get 3249%2F100.


d=sqrt%28361%2F10%29 Add 361%2F100 to 3249%2F100 to get 361%2F10.


d=6.008 Approximate the right side with a calculator


So the distance between the two points is about 6.008 units.


Rounding this value to the nearest hundredth gives us d=6.01


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Answer:


So the distance from the point A(-2,-2) to the line joining B(5,2) and C(-1.4) is approximately 6.01 units



Note: if you want more practice, check out this solver