Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s).

triangle abc is defined by the points \\(a(3,8)\\), \\(b(7,5)\\), and \\(c(2,3)\\).

create an equation for a line passing through point a and perpendicular to \\(\overline{bc}\\).

\\y = \text{_ } x + \text{_}\\

Explanation:

Find the slope of line segment BC

Using the Coordinate Geometry knowledge point, we calculate the slope \(m_{BC}\) of the line passing through \(B(7,5)\) and \(C(2,3)\).

$$ m_{BC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 5}{2 - 7} = \frac{-2}{-5} = \frac{2}{5} $$

Determine the perpendicular slope

Since the required line is perpendicular to \(BC\), its slope \(m\) must be the negative reciprocal of \(m_{BC}\).

$$ m = -\frac{1}{m_{BC}} = -\frac{1}{\frac{2}{5}} = -\frac{5}{2} $$

Write the equation in slope-intercept form

Using the point \(A(3,8)\) and the perpendicular slope \(m = -\frac{5}{2}\), we write the equation in point-slope form.

$$ y - y_1 = m(x - x_1) \implies y - 8 = -\frac{5}{2}(x - 3) $$

Now, we solve for \(y\) to convert it into slope-intercept form \(y = mx + b\).

$$ y - 8 = -\frac{5}{2}x + \frac{15}{2} \implies y = -\frac{5}{2}x + \frac{15}{2} + \frac{16}{2} \implies y = -\frac{5}{2}x + \frac{31}{2} $$

Answer:

Triangle ABC is defined by the points \(A(3,8)\), \(B(7,5)\), and \(C(2,3)\).
Create an equation for a line passing through point A and perpendicular to \(\overline{BC}\).

\(y =\) <blank>\(-5/2\)</blank> \(x +\) <blank>\(31/2\)</blank>