Question

graph \\(\delta rst\\) with vertices \\(r(4, 1)\\), \\(s(7, 3)\\), and \\(t(6, 4)\\) and its image after the glide reflection.

translation: \\((x, y) \
ightarrow (x - 3, y)\\)

reflection: in the line \\(y = -1\\)

Explanation:

Identify the vertices of the original triangle

We are given the original triangle \(\Delta RST\) with the following vertices:

• \(R(4, 1)\)
• \(S(7, 3)\)
• \(T(6, 4)\)

Apply the translation transformation

The first part of the glide reflection is the translation:

$$(x, y) ightarrow (x - 3, y)$$

Applying this rule to each vertex of \(\Delta RST\) to find the intermediate vertices \(R', S', T'\):

• \(R'(4 - 3, 1) = R'(1, 1)\)
• \(S'(7 - 3, 3) = S'(4, 3)\)
• \(T'(6 - 3, 4) = T'(3, 4)\)

Apply the reflection transformation

The second part of the glide reflection is the reflection in the line \(y = -1\).
For any point \((x, y)\), its reflection \((x, y'')\) across a horizontal line \(y = k\) is given by:

$$y'' = 2k - y$$

Here, \(k = -1\), so the reflection formula is:

$$(x, y) ightarrow (x, 2(-1) - y) = (x, -2 - y)$$

Applying this to the intermediate vertices \(R', S', T'\) to find the final image vertices \(R'', S'', T''\):

• For \(R'(1, 1)\):

$$R''(1, -2 - 1) = R''(1, -3)$$

• For \(S'(4, 3)\):

$$S''(4, -2 - 3) = S''(4, -5)$$

• For \(T'(3, 4)\):

$$T''(3, -2 - 4) = T''(3, -6)$$

Summarize the final coordinates

The final vertices of the image after the glide reflection are:

• \(R''(1, -3)\)
• \(S''(4, -5)\)
• \(T''(3, -6)\)

Answer:

The vertices of the original triangle and its final image after the glide reflection are:

Original Triangle \(\Delta RST\):

• \(R(4, 1)\)
• \(S(7, 3)\)
• \(T(6, 4)\)

Final Image \(\Delta R''S''T''\):

• \(R''(1, -3)\)
• \(S''(4, -5)\)
• \(T''(3, -6)\)