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!!!!

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) (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

.

Start with the slope formula.

Plug in , , , and

Subtract from to get

Subtract from to get

Reduce

So the slope of the line that goes through the points

and

is

Now let's use the point slope formula:

Start with the point slope formula

Plug in , , and

Distribute

Multiply

Add 2 to both sides.

Combine like terms.

Simplify

So the equation that goes through the points

and

is

--------------------------------------------------------------------------

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 is . Change the sign and flip the fraction to get or just

So the perpendicular slope is

--------------------------------------------------------------------------

Step 3) Find the equation of the line that has a slope of (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 which goes through the point (-2,-2) you can simply use the point-slope formula to find the equation:

---Point-Slope Formula---

where is the slope, and

is the given point

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

Plug in , , and (these values are given)

Rewrite as

Rewrite as

Distribute

Multiply and to get

Subtract 2 from both sides to isolate y

Combine like terms and to get

So the equation of the line that has a slope of goes through (-2,-2) is

--------------------------------------------------------------------------

Step 4) Find the point of intersection between the two lines and

Start with the second equation.

Plug in

Multiply EVERY term by the LCD 3 to clear out the fractions.

Subtract from both sides.

Subtract from both sides.

Combine like terms on the left side.

Combine like terms on the right side.

Divide both sides by to isolate .

Reduce.

Go back to the second equation.

Plug in

Multiply

Combine like terms.

So the lines intersect at the point

--------------------------------------------------------------------------

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

Start with the distance formula.

Plug in , , , and .

Subtract from to get .

Subtract from to get .

Square to get .

Square to get .

Add to to get .

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

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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
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