Question

1. using the digits 1 - 9 at most one time each, fill in the blanks so that the two lines are perpendicular.
2. line segment ab is shown on the coordinate plane below. line cd is the perpendicular bisector of line ab. a perpendicular bisector is perpendicular to the given line segment and intersects the line segment at its midpoint. what is the equation of line cd?

Explanation:

Step1: Find the mid - point of line segment AB

Let \(A(x_1,y_1)\) and \(B(x_2,y_2)\). From the graph, assume \(A(x_1,y_1)=(2,6)\) and \(B(x_2,y_2)=(8,2)\). The mid - point formula is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). So \(M(\frac{2 + 8}{2},\frac{6+2}{2})=M(5,4)\).

Step2: Find the slope of line segment AB

The slope formula is \(m_{AB}=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 2,y_1 = 6,x_2 = 8,y_2 = 2\), we get \(m_{AB}=\frac{2 - 6}{8 - 2}=\frac{-4}{6}=-\frac{2}{3}\).

Step3: Find the slope of line CD

Since line CD is perpendicular to line AB, the product of their slopes \(m_{AB}\times m_{CD}=- 1\). If \(m_{AB}=-\frac{2}{3}\), then \(m_{CD}=\frac{3}{2}\).

Step4: Find the equation of line CD

Using the point - slope form \(y - y_0=m(x - x_0)\), where \((x_0,y_0)=(5,4)\) and \(m = \frac{3}{2}\).
\(y - 4=\frac{3}{2}(x - 5)\)
\(y-4=\frac{3}{2}x-\frac{15}{2}\)
\(y=\frac{3}{2}x-\frac{15}{2}+4\)
\(y=\frac{3}{2}x-\frac{15 - 8}{2}\)
\(y=\frac{3}{2}x-\frac{7}{2}\)

Answer:

\(y=\frac{3}{2}x-\frac{7}{2}\)