Question

jessica is looking at a map of her property on the coordinate plane. there is a straight fence through the points (-2,7) and (6,-1). there is a tree at the point (-3,-6). find the distance between the tree and the fence. round your answer to the nearest tenth.
rounded distance =

Explanation:

Step1: Find the equation of the fence

The slope \(m\) of the line passing through \((x_1=-2,y_1 = 7)\) and \((x_2=6,y_2=-1)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1 - 7}{6-(-2)}=\frac{-8}{8}=-1\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((-2,7)\), we get \(y - 7=-1(x + 2)\), which simplifies to \(y=-x + 5\), or \(x+y-5 = 0\).

Step2: Use the distance formula from a point to a line

The distance \(d\) from the point \((x_0=-3,y_0=-6)\) to the line \(Ax+By + C=0\) (here \(A = 1\), \(B = 1\), \(C=-5\)) is given by the formula \(d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^{2}+B^{2}}}\).
Substitute \(x_0=-3\), \(y_0=-6\), \(A = 1\), \(B = 1\), \(C=-5\) into the formula: \(d=\frac{\vert1\times(-3)+1\times(-6)-5\vert}{\sqrt{1^{2}+1^{2}}}=\frac{\vert-3-6 - 5\vert}{\sqrt{2}}=\frac{\vert-14\vert}{\sqrt{2}}=\frac{14}{\sqrt{2}} = 7\sqrt{2}\approx9.9\).

Answer:

\(9.9\)