Question

find the image of (1, 2) after a reflection about x = 3 followed by a reflection about x = 7. (9, ?) enter the number that belongs in the green box.

Explanation:

Step1: Analyze reflection over vertical line

Reflecting a point \((x,y)\) over the vertical line \(x = a\) changes the \(x\)-coordinate, and the \(y\)-coordinate remains the same. The formula for reflection over \(x = a\) is \(x'=2a - x\), \(y' = y\). First, reflect \((1,2)\) over \(x = 3\). Here, \(a = 3\), \(x = 1\), \(y = 2\). So \(x_1=2\times3 - 1=5\), \(y_1 = 2\). The point after first reflection is \((5,2)\).

Step2: Reflect the new point over \(x = 7\)

Now reflect \((5,2)\) over \(x = 7\). Using the same formula, \(a = 7\), \(x = 5\), \(y = 2\). Then \(x_2=2\times7 - 5 = 9\), \(y_2 = 2\). The \(y\)-coordinate remains 2 throughout both reflections because reflections over vertical lines don't affect the \(y\)-coordinate.

Answer:

2