What is the distance between the line going through points (1, − 5 4 ) and ( 1 3 , − 3 4 ) and the point (2, −3)?

Answer 1

Mark's question is important and the question cannot be answered unless Mark's question is answered first!

However, as a reminder the distance from a point to a line is given by

(ax₁ + by₁ + c)/sqrt(a² + b²)

where ax + by + c is the general form of the given equation and(x₁,y₁) is the point in question.

Usually the absolute value of the expression is the distance.

Answer 2

I show each and every step to help those most in need, so I apologize to those bored with a highly-detailed solution.

First we need to find the equation of the line. The slope is found using any two points on the line in any order:

m=Δy/Δx=(y₂-y₁)/(x₂-x₁)

Let (x₂,y₂)=(1,-54) and (x₁,y₁)=(13,-34)

m=(-54--34)/(1-13)=(-54+34)/(-12)=-20/-12=5/3

Now use the slope-intercept form to get the equation:

y=mx+b

Substitute any point into that form. I choose (x₁,y₁)=(13,-34)

-34=(5/3)(13)+b

b=-34-65/3=-167/3

Finally, the equation of the line is y=(5/3)x-167/3.

Now the distance, d, from a point (x₀,y₀) to the line ax+by+c=0 is:

d=|ax₀+by₀+c|/[a²+b²]^1/2

(from https://brilliant.org/wiki/dot-product-distance-between-point-and-a-line/)

So we first have to write the equation of the line in this form:

y=(5/3)x-167/3

3(y)=3[(5/3)x-167/3]

3y=5x-167

-5x+3y+167=0

The point from which we want to find the distance from is (2,-3). So following the formula above:

d = |ax₀+by₀+c|/[a²+b²]^1/2

= |(-5)(2)+(3)(-3)+167|/[(-5)²+3²]^1/2

=|-10-9+167|/[25+9]^1/2

=|148|/34^1/2=148/34^1/2=25.38

but since the data were given with just two significant figures, we round it down to 25. The solution is 25.