Grade 7, Unit 1 - Practice Problems - Open Up Resources - Free Printable

Looking at the image, we are presented with an “Original Figure” — a stylized letter ‘A’ made of straight lines on a grid — and four other figures labeled Figure 1 through Figure 4. The task is to determine which of these figures is a dilation (also called a scaling or similarity transformation) of the original.

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🔍 What is a Dilation?

A dilation is a transformation that:
- Changes the size of a figure by a scale factor.
- Preserves the shape and angles (i.e., it’s similar to the original).
- Does not rotate, reflect, or distort the figure — only scales it uniformly from a center point.
- All corresponding lengths are multiplied by the same scale factor.
- Corresponding angles remain equal.

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🧩 Step-by-Step Analysis

Let’s analyze each figure relative to the Original Figure.

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#### ✔ Figure 1

- Looks identical in size and orientation to the Original Figure.
- This is a translation (shifted down and left), not a dilation.
- Since there’s no change in size, it’s technically a dilation with scale factor = 1, but usually “dilation” implies a *change* in size.
- However, mathematically, a scale factor of 1 *is* a dilation. But let’s check others to see if any have *different* scale factors.

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#### ✔ Figure 2

- Clearly larger than the original.
- Shape is preserved: same angles, same proportions.
- Let’s measure key dimensions:

- Original Figure:
- Height (from base to top vertex): 3 grid units.
- Width at base: 2 grid units.
- Horizontal bar inside: located 1 unit up from base, length ≈ 1 unit.

- Figure 2:
- Height: 6 grid units → 2× original.
- Width at base: 4 grid units → 2× original.
- Horizontal bar: located 2 units up from base, length ≈ 2 units → 2× original.

✔ So, Figure 2 is a dilation with scale factor 2.

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#### ✘ Figure 3

- Orientation is different — it’s rotated and possibly reflected.
- It’s also distorted: the angles are not preserved.
- For example, the original has two slanted sides meeting at a point; this one has a right angle at the bottom-left.
- This is not a dilation — it’s a completely different shape (looks like a right triangle with a horizontal bar).

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#### ✘ Figure 4

- Much smaller than original.
- But look closely: the proportions are off.
- Height: ~1.5 units? (hard to tell exactly, but clearly not scaled proportionally)
- The horizontal bar is almost as wide as the base — in the original, the bar is half the width of the base.
- In Figure 4, the bar appears nearly the same width as the base → distortion.

✔ So, Figure 4 is NOT a dilation — it’s either misproportioned or perhaps a poor drawing.

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🎯 Final Answer:

> Figure 2 is the only figure that is a true dilation of the original — it is scaled up by a factor of 2 while preserving all angles and proportions.

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💡 Bonus Note:

If the question allows for identity transformations (scale factor 1), then Figure 1 could also be considered a dilation. But typically, when asked to identify a dilation among options, they mean a *non-trivial* dilation (i.e., size changed). Since Figure 2 is clearly scaled and matches perfectly, it’s the intended answer.

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✔ Answer: Figure 2