Question

t → t(2,5) by a glide reflection where the translation is (x,y) → (x + 2,y) and the reflection line y = 1.
a. should t be to the left or to the right of t?
b. should t be above or below t?
c. what are the coordinates of t?

Explanation:

Step1: Analyze the translation part

The translation is \((x,y)\to(x + 2,y)\). This means that to get from the pre - image (point \(T\)) to the image after translation (before reflection), we move 2 units to the right. So, to reverse the translation (to get from \(T'\) back to the pre - reflection image), we move 2 units to the left.

Step2: Analyze the reflection part

The reflection line is \(y = 1\). Let the coordinates of the point after translation (before reflection) be \((x_1,y_1)\) and of \(T'\) be \((x_2,y_2)=(2,5)\). The formula for reflection over the line \(y = c\) is \((x,y)\to(x,2c - y)\). Here \(c = 1\), so if \((x_1,y_1)\) is reflected over \(y = 1\) to get \((x_2,y_2)\), then \(y_2=2\times1 - y_1\), so \(y_1=2 - y_2\).

Step3: Find the coordinates of the pre - reflection point

Since \(y_2 = 5\), then \(y_1=2-5=-3\). And since \(x_2=x_1\) (because reflection over a horizontal line does not change the \(x\) - coordinate), and \(x_2 = 2\) (from \(T'(2,5)\)), the pre - reflection point is \((2,-3)\).

Step4: Reverse the translation

To reverse the translation \((x,y)\to(x + 2,y)\) (i.e., to get from the pre - reflection point \((x_1,y_1)\) to \(T=(x,y)\)), we use the rule \((x_1,y_1)\to(x_1-2,y_1)\). Substituting \(x_1 = 2\) and \(y_1=-3\), we get \(T=(2 - 2,-3)=(0,-3)\)

Part a

Since the translation is \(x\to x + 2\) (right - ward), to reverse the translation (to get from \(T'\) back to \(T\)), \(T\) is to the left of \(T'\)

Part b

The \(y\) - coordinate of \(T\) is \(-3\) and of \(T'\) is \(5\). So \(T\) is below \(T'\)

Part c

We found that \(T=(0,-3)\)

Answer:

a. To the left of \(T'\)
b. Below \(T'\)
c. \((0,-3)\)