Question

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

bc reflects about a line such that n is the reflection of b and o is the reflection of c. point n is shown on the coordinate plane, but point o is not.

the coordinates of point o are (□,□).

Explanation:

Step1: Find the reflection line

Since \(N\) is the reflection of \(B\), and \(B=(3,7)\), \(N=(3,5)\). The \(x -\)coordinate remains the same. The mid - point of \(B\) and \(N\) gives the equation of the reflection line. The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For \(B(3,7)\) and \(N(3,5)\), the mid - point is \((3,\frac{7 + 5}{2})=(3,6)\). The reflection line is \(y = 6\).

Step2: Use the reflection property for point \(C\)

Point \(C=(6,7)\). Let the coordinates of \(O\) be \((x,y)\). Since the reflection is about the line \(y = 6\), the \(x -\)coordinate of \(O\) is the same as the \(x -\)coordinate of \(C\) (because reflection about a horizontal line \(y = k\) does not change the \(x -\)coordinate). For the \(y -\)coordinate, use the formula \(y=2k - y_1\), where \(k = 6\) and \(y_1 = 7\). So \(y=2\times6-7\).

$$y = 12 - 7=5$$

Answer:

\((6,5)\)