what is the equation of the line that is perpendicular to the given line and passes through the point (5…

Explanation:

Step1: Find the slope of the given line

First, find two - points on the given line. Let's take the points ((- 8,10)) and ((8,-10)). The slope formula is (m=\frac{y_2 - y_1}{x_2 - x_1}). So, (m_1=\frac{-10 - 10}{8+8}=\frac{-20}{16}=-\frac{5}{4}).

Step2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is (- 1), i.e., (m_1\times m_2=-1). Given (m_1 =-\frac{5}{4}), then (-\frac{5}{4}\times m_2=-1), so (m_2=\frac{4}{5}).

Step3: Use the point - slope form to find the equation of the line

The point - slope form of a line is (y - y_1=m(x - x_1)), where ((x_1,y_1)=(5,3)) and (m = \frac{4}{5}). So (y - 3=\frac{4}{5}(x - 5)).

Step4: Convert to standard form

Expand the point - slope form: (y-3=\frac{4}{5}x - 4). Multiply through by 5 to get (5y-15 = 4x-20). Rearrange to (4x-5y = 5).

Answer:

A. (4x - 5y=5)