Math worksheet for practicing dilations on a coordinate grid, including problems with scale factors and graphing exercises.

Let’s solve each problem step by step.

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Problem 1, Part 1: Dilate figure ABCD with scale factor 1/3

Original points:
- A (6, -4)
- B (9, -4)
- C (9, -9)
- D (6, -9)

To dilate from the origin (0,0), multiply each x and y coordinate by the scale factor (1/3).

A’ = (6 × 1/3, -4 × 1/3) = (2, -4/3) → but we usually keep as fractions or decimals? Let’s use fractions for accuracy.

Actually, let’s write them as mixed numbers or improper fractions — but since it’s a grid, maybe they expect decimal or fraction? The worksheet doesn’t specify, so we’ll use exact fractions.

But wait — looking at the graph, the original figure is in quadrant IV. After dilation by 1/3, it should be closer to origin.

Let’s compute:

A’: (6 * 1/3, -4 * 1/3) = (2, -4/3)
B’: (9 * 1/3, -4 * 1/3) = (3, -4/3)
C’: (9 * 1/3, -9 * 1/3) = (3, -3)
D’: (6 * 1/3, -9 * 1/3) = (2, -3)

So new coordinates:
A’ (2, -4/3), B’ (3, -4/3), C’ (3, -3), D’ (2, -3)

But -4/3 is approximately -1.333… which might be hard to plot exactly on grid. Maybe the problem expects us to leave as fractions? Or perhaps I made a mistake?

Wait — let me double-check the original points. The figure ABCD is a rectangle. From A(6,-4) to B(9,-4): that’s 3 units right. From B(9,-4) to C(9,-9): 5 units down. So yes, correct.

After dilation by 1/3:
Width becomes 3 * 1/3 = 1 unit
Height becomes 5 * 1/3 ≈ 1.666... units

So coordinates are correct.

But perhaps the worksheet expects decimal? Or maybe I should write as improper fractions? Let’s keep as fractions for now.

Alternatively, maybe the center of dilation is not origin? But the problem says “dilate” without specifying center — in coordinate grid problems like this, unless stated otherwise, we assume center is origin (0,0).

So we proceed.

Final for part 1:
A’ (2, -4/3), B’ (3, -4/3), C’ (3, -3), D’ (2, -3)

But let’s check if we can write -4/3 as -1 1/3 — but for listing coordinates, better to use improper fractions or decimals? Since the original were integers, and scale factor is 1/3, fractions are acceptable.

However, looking ahead — other problems have integer results. Maybe I miscalculated?

Wait — point A is (6, -4). 6 * 1/3 = 2, good. -4 * 1/3 = -4/3. That’s correct.

Perhaps the problem allows fractional coordinates. We’ll go with that.

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Problem 1, Part 2: Dilate figure ABC with scale factor 4

Points given:
A (-1, 2)
B (1, 2)
C (1, -2)
D (-2, -1) ← Wait, the problem says "figure ABC", but lists D too? Let me read again.

It says: “Graph figure ABC after a dilation with a scale factor of 4.” Then lists A, B, C, D. Probably typo — maybe it’s quadrilateral ABCD? Because D is listed and shown in graph.

Looking at the graph description: there’s a small rectangle near origin labeled A,B,C,D. So likely it’s ABCD.

So points:
A (-1, 2)
B (1, 2)
C (1, -2)
D (-2, -1) ← Wait, D is (-2,-1)? But in the graph, if A(-1,2), B(1,2), C(1,-2), then D should be (-1,-2) to make rectangle? But here D is given as (-2,-1). That would make it not a rectangle? Let me check.

From A(-1,2) to B(1,2): horizontal, length 2.
B(1,2) to C(1,-2): vertical, length 4.
C(1,-2) to D(-2,-1): that’s left 3, up 1 — not perpendicular? Hmm.

Perhaps it’s a different shape. Anyway, we take the points as given.

So dilate each by scale factor 4 from origin.

A’ = (-1 * 4, 2 * 4) = (-4, 8)
B’ = (1 * 4, 2 * 4) = (4, 8)
C’ = (1 * 4, -2 * 4) = (4, -8)
D’ = (-2 * 4, -1 * 4) = (-8, -4)

So new coordinates:
A’ (-4, 8), B’ (4, 8), C’ (4, -8), D’ (-8, -4)

That seems straightforward.

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Problem 2, Part 1: Graph figure, then dilate with scale factor 3

Points:
A (-3, -1)
B (-3, -3)
C (-3, -2) ← Wait, all have x=-3? That means it’s a vertical line segment? But three points? A(-3,-1), B(-3,-3), C(-3,-2) — so C is between A and B? That would be colinear. Probably a triangle? But all same x-coordinate — degenerate triangle? Or maybe typo?

Wait, let me read: “Graph the figure, then dilate with a scale factor of 3.” Points A(-3,-1), B(-3,-3), C(-3,-2). All x=-3, so it’s a vertical line from y=-3 to y=-1, with C at y=-2. So it’s just a line segment, but with an extra point? Perhaps it’s meant to be a triangle with another point? Or maybe it’s a mistake.

Looking back at the worksheet image description — but we can't see image. Based on text, we have to go with what's written.

Perhaps C is supposed to be different? But as per given, we proceed.

Dilate each by 3:

A’ = (-3*3, -1*3) = (-9, -3)
B’ = (-3*3, -3*3) = (-9, -9)
C’ = (-3*3, -2*3) = (-9, -6)

So all still on x=-9 line. New coordinates: A’(-9,-3), B’(-9,-9), C’(-9,-6)

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Problem 2, Part 2: Graph figure, then dilate with scale factor 1/4

Points:
A (-8, 4)
B (-8, 8)
C (-4, 8)
D (-4, 4)

This looks like a rectangle: from x=-8 to -4 (width 4), y=4 to 8 (height 4). Square actually.

Dilate by 1/4:

A’ = (-8 * 1/4, 4 * 1/4) = (-2, 1)
B’ = (-8 * 1/4, 8 * 1/4) = (-2, 2)
C’ = (-4 * 1/4, 8 * 1/4) = (-1, 2)
D’ = (-4 * 1/4, 4 * 1/4) = (-1, 1)

Perfect — all integers.

New coordinates: A’(-2,1), B’(-2,2), C’(-1,2), D’(-1,1)

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Now, compiling all answers.

For Problem 1, Part 1: We have fractional coordinates. But let me confirm if scale factor applies correctly.

Original A(6,-4), scale 1/3: (2, -4/3). Yes.

But perhaps in some curricula, they expect to round or something? No, better to keep exact.

Maybe write as decimals? -4/3 ≈ -1.333, but not nice.

I think fractions are fine.

Similarly, others are integers.

So final answers:

Practice Problems 1:

1) After dilation with scale factor 1/3:
A’ (2, -4/3), B’ (3, -4/3), C’ (3, -3), D’ (2, -3)

2) After dilation with scale factor 4:
A’ (-4, 8), B’ (4, 8), C’ (4, -8), D’ (-8, -4)

Practice Problems 2:

1) After dilation with scale factor 3:
A’ (-9, -3), B’ (-9, -9), C’ (-9, -6)

2) After dilation with scale factor 1/4:
A’ (-2, 1), B’ (-2, 2), C’ (-1, 2), D’ (-1, 1)

Note: In Problem 2 Part 1, the figure is degenerate (all points colinear), but we followed instructions.

Also, in Problem 1 Part 2, even though D was listed under "figure ABC", we included it since it was provided and likely part of the figure.

Now, for Final Answer section, we need to list all new coordinates as requested.

The instruction says: “list the location of the new coordinates” for each.

So we present them clearly.

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Final Answer:

Practice Problems 1:

1) Dilated coordinates with scale factor 1/3:
A' (2, -4/3), B' (3, -4/3), C' (3, -3), D' (2, -3)

2) Dilated coordinates with scale factor 4:
A' (-4, 8), B' (4, 8), C' (4, -8), D' (-8, -4)

Practice Problems 2:

1) Dilated coordinates with scale factor 3:
A' (-9, -3), B' (-9, -9), C' (-9, -6)

2) Dilated coordinates with scale factor 1/4:
A' (-2, 1), B' (-2, 2), C' (-1, 2), D' (-1, 1)