Worksheet for learning translations in geometry with labeled exercises and coordinate grids.

Problem Analysis:

The task involves performing translations on geometric figures and labeling the resulting images. Let's break it down step by step.

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#### Part 1: Translating a Triangle
The first part of the problem asks us to sketch the image of a pre-image after a given translation. The translation is described as "Slide 6 right and 5 down." This means:
- Every point in the figure will move 6 units to the right.
- Every point in the figure will move 5 units downward.

##### Steps:
1. Identify the vertices of the triangle in the pre-image.
2. Apply the translation vector \((+6, -5)\) to each vertex.
3. Plot the new vertices and connect them to form the translated triangle.
4. Label the new vertices as \(A'\), \(B'\), \(C'\), etc.

##### Solution:
- Suppose the original vertices of the triangle are:
- \(A = (x_1, y_1)\)
- \(B = (x_2, y_2)\)
- \(C = (x_3, y_3)\)

- After the translation \((+6, -5)\):
- \(A' = (x_1 + 6, y_1 - 5)\)
- \(B' = (x_2 + 6, y_2 - 5)\)
- \(C' = (x_3 + 6, y_3 - 5)\)

- Sketch the new triangle using these coordinates and label the vertices as \(A'\), \(B'\), and \(C'\).

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#### Part 2: Translating a Quadrilateral
The second part of the problem involves translating a quadrilateral defined by specific coordinates. The translation vector is given as \((8, -6)\). This means:
- Every point in the figure will move 8 units to the right.
- Every point in the figure will move 6 units downward.

##### Steps:
1. Identify the vertices of the quadrilateral in the pre-image.
2. Apply the translation vector \((+8, -6)\) to each vertex.
3. Plot the new vertices and connect them to form the translated quadrilateral.
4. Label the new vertices as necessary.

##### Solution:
- Suppose the original vertices of the quadrilateral are:
- \(D = (x_1, y_1)\)
- \(E = (x_2, y_2)\)
- \(F = (x_3, y_3)\)
- \(G = (x_4, y_4)\)

- After the translation \((+8, -6)\):
- \(D' = (x_1 + 8, y_1 - 6)\)
- \(E' = (x_2 + 8, y_2 - 6)\)
- \(F' = (x_3 + 8, y_3 - 6)\)
- \(G' = (x_4 + 8, y_4 - 6)\)

- Sketch the new quadrilateral using these coordinates and label the vertices as \(D'\), \(E'\), \(F'\), and \(G'\).

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#### Part 3: Graphing and Transforming a Polygon
The third part of the problem involves graphing a polygon defined by specific coordinates and then applying a transformation rule. The coordinates of the polygon are:
- \(P(4, 8)\)
- \(Q(2, 6)\)
- \(R(2, 3)\)
- \(S(5, 7)\)

The transformation rule is:
\[
(x, y) \rightarrow (x + 6, y - 3)
\]

##### Steps:
1. Plot the original polygon using the given coordinates.
2. Apply the transformation rule \((x + 6, y - 3)\) to each vertex.
3. Plot the new vertices and connect them to form the transformed polygon.
4. Label the new vertices as necessary.

##### Solution:
- Original vertices:
- \(P(4, 8)\)
- \(Q(2, 6)\)
- \(R(2, 3)\)
- \(S(5, 7)\)

- After the transformation \((x + 6, y - 3)\):
- \(P' = (4 + 6, 8 - 3) = (10, 5)\)
- \(Q' = (2 + 6, 6 - 3) = (8, 3)\)
- \(R' = (2 + 6, 3 - 3) = (8, 0)\)
- \(S' = (5 + 6, 7 - 3) = (11, 4)\)

- Sketch the new polygon using these coordinates and label the vertices as \(P'\), \(Q'\), \(R'\), and \(S'\).

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Final Answer:

\[
\boxed{\text{See detailed steps above for each part.}}
\]