Question

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completes the statement or answers the qu
COORDINATE GEOMETRY Find the distance from P to
1. Line / contains points (3,5) and (7,9). Point P has coordinates
a. 32
b.18
c. /20
d. 26​

Answer 1

`d=3√(2)\ units`

Explanation:

The complete question is

Line L contains points (3, 5) and (7, 9). Points P has coordinates (2, 10). Find the distance from P to L

step 1

Find the equation of the line L contains points (3,5) and (7,9)

Find the slope

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

substitute the given values

`m=(9-5)/(7-3)`

`m=(4)/(4)=1`

Find the equation of the line in point slope form

`y-y1=m(x-x1)`

we have

`m=1\\point\ (3,5)`

substitute

`y-5=(1)(x-3)`

isolate the variable y

`y=x-3+5`

`y=x+2` -----> equation A

step 2

Find the equation of the perpendicular line to the given line L that passes through the point P

Remember that

If two lines are perpendicular, then their slopes are opposite reciprocal

so

The slope of the given line L is
`m=1`

The slope of the line perpendicular to the given line L is

`m=-1`

Find the equation of the line in point slope form

`y-y1=m(x-x1)`

we have

`m=-1\\point\ (2,10)`

substitute

`y-10=-(x-2)`

isolate the variable y

`y=-x+2+10`

`y=-x+12` ----> equation B

step 3

Find the intersection point equation A and equation B

`y=x+2` -----> equation A

`y=-x+12` ----> equation B

solve the system by graphing

The intersection point is (5.7)

see the attached figure

step 4

we know that

The distance from point P to the the line L is equal to the distance between the point P and point (5,7)

the formula to calculate the distance between two points is equal to

`d=\sqrt{(y2-y1)^(2)+(x2-x1)^(2)}`

we have

(2,10) and (5,7)

substitute

`d=\sqrt{(7-10)^(2)+(5-2)^(2)}`

`d=\sqrt{(-3)^(2)+(3)^(2)}`

`d=√(18)\ units`

simplify

`d=3√(2)\ units`

see the attached figure to better understand the problem