Answer:
![d=3\sqrt{2}\ units](https://tex.z-dn.net/?f=d%3D3%5Csqrt%7B2%7D%5C%20units)
Step-by-step explanation:
<u><em>The complete question is</em></u>
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)
<em>Find the slope</em>
The formula to calculate the slope between two points is equal to
![m=\frac{y2-y1}{x2-x1}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By2-y1%7D%7Bx2-x1%7D)
substitute the given values
![m=\frac{9-5}{7-3}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B9-5%7D%7B7-3%7D)
![m=\frac{4}{4}=1](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B4%7D%7B4%7D%3D1)
<em>Find the equation of the line in point slope form</em>
![y-y1=m(x-x1)](https://tex.z-dn.net/?f=y-y1%3Dm%28x-x1%29)
we have
![m=1\\point\ (3,5)](https://tex.z-dn.net/?f=m%3D1%5C%5Cpoint%5C%20%283%2C5%29)
substitute
![y-5=(1)(x-3)](https://tex.z-dn.net/?f=y-5%3D%281%29%28x-3%29)
isolate the variable y
![y=x-3+5](https://tex.z-dn.net/?f=y%3Dx-3%2B5)
-----> 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](https://tex.z-dn.net/?f=m%3D1)
The slope of the line perpendicular to the given line L is
![m=-1](https://tex.z-dn.net/?f=m%3D-1)
<em>Find the equation of the line in point slope form</em>
![y-y1=m(x-x1)](https://tex.z-dn.net/?f=y-y1%3Dm%28x-x1%29)
we have
![m=-1\\point\ (2,10)](https://tex.z-dn.net/?f=m%3D-1%5C%5Cpoint%5C%20%282%2C10%29)
substitute
![y-10=-(x-2)](https://tex.z-dn.net/?f=y-10%3D-%28x-2%29)
isolate the variable y
![y=-x+2+10](https://tex.z-dn.net/?f=y%3D-x%2B2%2B10)
----> equation B
step 3
Find the intersection point equation A and equation B
-----> equation A
----> 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}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28y2-y1%29%5E%7B2%7D%2B%28x2-x1%29%5E%7B2%7D%7D)
we have
(2,10) and (5,7)
substitute
![d=\sqrt{(7-10)^{2}+(5-2)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%287-10%29%5E%7B2%7D%2B%285-2%29%5E%7B2%7D%7D)
![d=\sqrt{(-3)^{2}+(3)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-3%29%5E%7B2%7D%2B%283%29%5E%7B2%7D%7D)
![d=\sqrt{18}\ units](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B18%7D%5C%20units)
simplify
![d=3\sqrt{2}\ units](https://tex.z-dn.net/?f=d%3D3%5Csqrt%7B2%7D%5C%20units)
see the attached figure to better understand the problem