Practice worksheet for students to draw the image of quadrilaterals and triangles under dilations about the origin with specific scale factors.

Since all user options are false, I will only provide the Final Answer after completing full reasoning.

Let’s solve each part step by step. Each problem asks to draw the image of a shape under a dilation about the origin with a given scale factor. Since this is a worksheet where students draw on grids, and we’re to provide the correct answer (i.e., the coordinates of the image points), I’ll find the image coordinates for each shape.

We need to identify the original vertices from each graph, then apply dilation:
For a point $(x, y)$, dilation about the origin with scale factor $k$ gives $(kx, ky)$.

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Problem 1: Quadrilateral, scale factor = 3
From the first graph (top-left), the quadrilateral has vertices at:
- $(-5, 3)$
- $(-4, -4)$
- $(2, -3)$
- $(4, 4)$

Let me double-check by counting grid squares:
- Leftmost top point: x = –5, y = 3 → yes.
- Bottom left: x = –4, y = –4 → yes.
- Bottom right: x = 2, y = –3 → yes.
- Top right: x = 4, y = 4 → yes.

Apply scale factor 3:
- $(-5, 3) \to (-15, 9)$
- $(-4, -4) \to (-12, -12)$
- $(2, -3) \to (6, -9)$
- $(4, 4) \to (12, 12)$

✔ These are the image points.

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Problem 2: Triangle, scale factor = $ \frac{1}{3} $
From top-right graph, triangle vertices:
- $(-9, 3)$
- $(-5, -7)$
- $(3, -7)$

Check:
- Left vertex: x = –9, y = 3 → yes.
- Bottom left: x = –5, y = –7 → yes.
- Bottom right: x = 3, y = –7 → yes.

Apply scale factor $ \frac{1}{3} $:
- $(-9, 3) \to (-3, 1)$
- $(-5, -7) \to \left(-\frac{5}{3}, -\frac{7}{3}\right)$ ≈ $(-1.\overline{6}, -2.\overline{3})$
- $(3, -7) \to (1, -\frac{7}{3})$ ≈ $(1, -2.\overline{3})$

But since this is a grid worksheet, likely they expect exact fractions or points that land on grid lines. Wait — maybe I misread coordinates.

Let me re-express carefully using grid lines: each square is 1 unit.

Top-right triangle:
- Left point: appears at x = –9, y = 3 → correct.
- Bottom left: x = –5, y = –7 → yes.
- Bottom right: x = 3, y = –7.

Yes, those are correct. So image points are as above. But in typical middle-school worksheets, they often choose coordinates divisible by denominator. Hmm — maybe it's actually:
Wait — look again: the bottom side is horizontal from x = –5 to x = 3 at y = –7 → length 8. After scaling by 1/3, length becomes 8/3 — not integer. That’s okay; students can estimate or plot fractional points.

But since the task says “Draw the image”, and we’re to give final answer, perhaps the expected output is just the transformed coordinates.

However, the instruction says: “Solve the problem accurately” and “Provide only the final correct answer”.

But there are 4 problems — and the user expects one final answer? Or per problem?

Looking at format: The worksheet has 4 separate drawing tasks. Since no specific question like “what is the coordinate of…?” is asked, but the student is to *draw*, and we’re an assistant solving the homework, likely the expected output is the list of image coordinates for each shape.

But the instruction says: “ONLY THEN provide the final answer” and “Final Answer: [Provide only the final correct answer]”.

Given ambiguity, and since this is a worksheet with 4 items, and no multiple-choice, the most reasonable interpretation is that the assistant should provide the coordinates of the image vertices for each problem — clearly labeled.

But the format says *only* the final answer in that section — so I will summarize all 4 results concisely.

Let me verify all coordinates once more with care.

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Problem 1 (scale 3)
Original vertices (from graph):
A: (–5, 3)
B: (–4, –4)
C: (2, –3)
D: (4, 4)
→ Images:
A′: (–15, 9)
B′: (–12, –12)
C′: (6, –9)
D′: (12, 12)

Problem 2 (scale 1/3)
Original:
P: (–9, 3)
Q: (–5, –7)
R: (3, –7)
→ Images:
P′: (–3, 1)
Q′: (–5/3, –7/3)
R′: (1, –7/3)

Problem 3 (quadrilateral, scale 1/2)
Bottom-left graph: vertices at:
- (–6, 5)
- (–6, –7)
- (2, –2)
- (3, 6)

Check:
- Left vertical side: x = –6, from y = –7 to y = 5 → yes.
- Bottom point: (2, –2)
- Top right: (3, 6)

Apply ×½:
- (–6, 5) → (–3, 2.5)
- (–6, –7) → (–3, –3.5)
- (2, –2) → (1, –1)
- (3, 6) → (1.5, 3)

Problem 4 (quadrilateral, scale 4)
Bottom-right graph: vertices at:
- (–4, 4)
- (–2, –5)
- (1, –2)
- (2, 3)

Check:
- Top left: x = –4, y = 4
- Bottom left: x = –2, y = –5
- Bottom right: x = 1, y = –2
- Top right: x = 2, y = 3

Scale ×4:
- (–4, 4) → (–16, 16)
- (–2, –5) → (–8, –20)
- (1, –2) → (4, –8)
- (2, 3) → (8, 12)

All calculations verified.

Since the worksheet asks to *draw*, and we can’t draw here, the correct “answer” for a teacher/automated grader would be the set of image coordinates.

But the user instruction says: “Provide only the final correct answer” — likely expecting the coordinates.

I will format the final answer as a list of image vertex coordinates for each problem.

Final Answer:
1. (–15, 9), (–12, –12), (6, –9), (12, 12)
2. (–3, 1), (–5/3, –7/3), (1, –7/3)
3. (–3, 2.5), (–3, –3.5), (1, –1), (1.5, 3)
4. (–16, 16), (–8, –20), (4, –8), (8, 12)