Question

question 8 (2 points)
find the distance from point e to \overleftrightarrow{fh}. round your answer to the nearest tenth, if necessary.
the distance from point e to \overleftrightarrow{fh} is \underline{\quad\quad\quad} units.

Explanation:

Step1: Find the equation of line \( \overleftrightarrow{FH} \)

The slope \( m \) of the line passing through \( F(0,3) \) and \( H(2,1) \) is \( m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{1 - 3}{2-0}=\frac{-2}{2}=-1 \).
Using the point - slope form \( y - y_{1}=m(x - x_{1}) \) with the point \( F(0,3) \), the equation of the line is \( y-3=-1(x - 0) \), which simplifies to \( y=-x + 3 \) or \( x+y-3 = 0 \).

Step2: Use the distance formula from a point \( (x_{0},y_{0}) \) to a line \( Ax+By+C = 0 \)

The distance \( d \) from a point \( E(-4,-3) \) (where \( x_{0}=-4,y_{0}=-3 \)) to the line \( x+y-3 = 0 \) (where \( A = 1,B = 1,C=-3 \)) is given by the formula \( d=\frac{\vert Ax_{0}+By_{0}+C\vert}{\sqrt{A^{2}+B^{2}}} \).
Substitute the values: \( d=\frac{\vert1\times(-4)+1\times(-3)-3\vert}{\sqrt{1^{2}+1^{2}}}=\frac{\vert-4 - 3-3\vert}{\sqrt{2}}=\frac{\vert-10\vert}{\sqrt{2}}=\frac{10}{\sqrt{2}} = 5\sqrt{2}\approx7.1 \).

Answer:

\( 7.1 \)