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 bc.

Explanation:

Step1: Find the slope of line BC

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $B(7,5)$ and $C(2,3)$, we have $m_{BC}=\frac{5 - 3}{7 - 2}=\frac{2}{5}$.

Step2: Find the slope of the line perpendicular to BC

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the line perpendicular to BC be $m_1$. Then $m_1\times m_{BC}=-1$. So $m_1=-\frac{5}{2}$.

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

The point - slope form of a line is $y - y_1=m(x - x_1)$. We use point $A(3,8)$ and $m =-\frac{5}{2}$. Substituting $x_1 = 3,y_1 = 8,m=-\frac{5}{2}$ into the point - slope form, we get $y - 8=-\frac{5}{2}(x - 3)$.

Step4: Convert to slope - intercept form

Expand the right - hand side: $y - 8=-\frac{5}{2}x+\frac{15}{2}$. Then add 8 to both sides: $y=-\frac{5}{2}x+\frac{15}{2}+8=-\frac{5}{2}x+\frac{15 + 16}{2}=-\frac{5}{2}x+\frac{31}{2}$.

Answer:

$y=-\frac{5}{2}x+\frac{31}{2}$