Determine the Scale Factor Between Two Shapes and Determine the ... - Free Printable

Let’s go through each pair of shapes one by one. We’ll find the scale factor first (how much bigger or smaller the second shape is compared to the first), then use that to find any missing lengths.

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First Pair: Rectangles A and A’

Rectangle A has:
- Length = 18.2 ft
- Width = 3.60 ft

Rectangle A’ has:
- Length = 9.1 ft
- Width = ? (but we can calculate it)

To find the scale factor, divide a side of A’ by the matching side of A.

Use length:
9.1 ÷ 18.2 = 0.5 → So scale factor is 0.5×

Now check width:
3.60 × 0.5 = 1.80 → which matches the given width in A’. Good!

So for this pair:
✔ Scale factor = 0.5×
✔ Missing width in A’ was already given as 1.80 — no new missing length here.

Wait — actually, looking again, both dimensions are labeled in A’, so maybe nothing is missing? But the problem says “label the missing lengths” — perhaps they want us to confirm or write them? In this case, all are given. Maybe just note the scale factor.

But let’s move on — maybe other pairs have actual missing values.

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Second Pair: Rectangles B and B’

Rectangle B:
- Height = 25.6 yd
- Width = 8.8 yd

Rectangle B’:
- Height = 38.4 yd
- Width = 13.2 yd

Find scale factor using height:
38.4 ÷ 25.6 = ?

Let’s compute:
25.6 × 1.5 = 38.4 → Yes! So scale factor = 1.5×

Check with width:
8.8 × 1.5 = 13.2 → Correct!

So scale factor = 1.5×

No missing lengths here either — all are labeled.

Hmm… maybe the red numbers are the answers? Let’s look at the third pair.

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Third Pair: Triangles C and C’

Triangle C:
- Base = 7 in
- Height = 10.5 in
- Hypotenuse = 12.6 in

Triangle C’:
- Base = 17.5 in
- Height = 26.25 in
- Hypotenuse = 31.5 in

Find scale factor using base:
17.5 ÷ 7 = 2.5 → So scale factor = 2.5×

Check height:
10.5 × 2.5 = 26.25 → Correct!

Check hypotenuse:
12.6 × 2.5 = 31.5 → Correct!

So scale factor = 2.5×

Again, all sides are labeled — no missing lengths?

Wait — maybe I misread. Let me check the last pair.

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Fourth Pair: Triangles D and D’

Triangle D:
- Base = 16.8 in
- Height = 4.8 in
- Hypotenuse = 17.5 in

Triangle D’:
- Base = 58.8 in
- Height = 16.8 in
- Hypotenuse = 61.25 in

Find scale factor using base:
58.8 ÷ 16.8 = ?

Let’s calculate:
16.8 × 3.5 = 58.8 → Yes! Because 16.8 × 3 = 50.4, plus 16.8 × 0.5 = 8.4 → total 58.8.

So scale factor = 3.5×

Check height:
4.8 × 3.5 = 16.8 → Correct!

Check hypotenuse:
17.5 × 3.5 = ?

17.5 × 3 = 52.5
17.5 × 0.5 = 8.75
Total = 52.5 + 8.75 = 61.25 → Correct!

So scale factor = 3.5×

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Wait — but the problem says “label the missing lengths”. In all cases, all lengths are already labeled in the image. Maybe the red numbers are the answers they want us to write? Or perhaps in the original worksheet, some were blank and here they’re filled?

Looking back at the user’s image description — it says “Answers” at the top, and red numbers are shown next to each pair: 0.5×, 1.5×, 2.5×, 3.5× — those are the scale factors.

And for missing lengths — in rectangle A’, the width is written as 1.80 in red — which is correct (3.60 × 0.5). Similarly, in others, all sides match when multiplied by the scale factor.

So perhaps the task was to find the scale factor and verify/fill in missing lengths — and in this answer key, everything is already done.

But since the student is asking to solve it, maybe we should present the scale factors and confirm the missing lengths if any.

Actually — re-examining the first rectangle pair:

In rectangle A’, the width is labeled as “1.80” in red — meaning it was probably blank originally, and now filled. Same for others? But in the image provided, all are filled.

Perhaps for the purpose of this exercise, we just need to state the scale factor for each pair, and if there was a missing length, calculate it.

Since in all cases, the scale factor leads to the given numbers, and no length is truly missing in the provided image, I think the main answer expected is the scale factors.

But let’s double-check if any length is actually missing.

Looking at Rectangle A’:

It shows:
- Length: 9.1 ft
- Width: 1.80 ft (in red) — so likely that was the missing one.

Similarly, in Triangle C’, all sides are given — but maybe in the original, one was blank? Not sure.

Given the instructions, and since the red numbers are the answers, I think we are to report the scale factors and the calculated missing lengths where applicable.

But in this case, for each pair, only the scale factor is needed as the "answer", and the missing lengths are derived from it.

Let me summarize:

Pair 1 (A to A’):
Scale factor = 0.5
Missing width in A’ = 3.60 × 0.5 = 1.80 ft → already given

Pair 2 (B to B’):
Scale factor = 1.5
All sides given — no missing

Pair 3 (C to C’):
Scale factor = 2.5
All sides given

Pair 4 (D to D’):
Scale factor = 3.5
All sides given

Wait — perhaps in the original problem, for example, in triangle C’, the hypotenuse was missing? But in the image, it's given as 31.5.

I think the safest approach is to provide the scale factor for each pair, and if a length was missing, show how to calculate it. Since in the answer key, the red numbers include both scale factors and some lengths, but for consistency, I'll list the scale factors as the primary answer, and note any missing length calculation.

But looking at the very first pair: Rectangle A’ has width labeled 1.80 in red — which is not in black like the others — so likely that was the missing length.

Similarly, in other pairs, all lengths are in black except the scale factors in red.

In Rectangle B’, all lengths are in black — 38.4 and 13.2 — and scale factor 1.5 in red.

In Triangle C’, all sides in black, scale factor 2.5 in red.

In Triangle D’, all sides in black, scale factor 3.5 in red.

Only in Rectangle A’, the width is in red — 1.80 — suggesting it was missing.

So perhaps for Pair 1, we need to calculate the missing width.

Let me do that clearly.

For Pair 1:

Shape A: width = 3.60 ft
Scale factor to A’ = 0.5 (from length: 9.1 / 18.2 = 0.5)
So missing width in A’ = 3.60 × 0.5 = 1.80 ft

For other pairs, no missing lengths — only scale factors to report.

But the problem says "for each pair of shapes and label the missing lengths" — implying there might be missing lengths in each, but in this answer key, only one is visibly missing (the width in A’).

Perhaps in the original worksheet, more were blank, but here they're filled.

To be thorough, I'll assume that for each pair, we state the scale factor, and if a length is missing, calculate it. Since only one is obviously missing (width of A’), I'll focus on that.

But let's check Triangle C’ — is any side missing? No, all given.

Another thought: in Rectangle B’, the height is 38.4, which is 25.6 × 1.5, and width 13.2 = 8.8 × 1.5 — all good.

Perhaps the "missing lengths" refer to the fact that in some cases, you have to calculate a side using the scale factor, even if it's given — but that doesn't make sense.

I think the best course is to provide the scale factor for each pair, and for the first pair, explicitly calculate the missing width.

So final answers per pair:

1. Rectangles A and A’:
Scale factor = 0.5
Missing width in A’ = 3.60 × 0.5 = 1.80 ft

2. Rectangles B and B’:
Scale factor = 1.5
No missing lengths (all given)

3. Triangles C and C’:
Scale factor = 2.5
No missing lengths

4. Triangles D and D’:
Scale factor = 3.5
No missing lengths

But since the problem likely expects the scale factors as the main answer, and the missing length for the first pair, I'll box the scale factors and the calculated missing length.

However, looking at the red text in the image, it shows:

- For A/A’: 0.5× and 1.80 (which is the width)
- For B/B’: 1.5×
- For C/C’: 2.5×
- For D/D’: 3.5×

So probably, the answers are the scale factors, and for the first pair, also the missing width.

But to match what's expected, I'll list the scale factors for each pair, and mention the missing length for the first pair.

Since the instruction is to "determine the scale factor for each pair... and label the missing lengths", I should provide both.

Let me write it out clearly.

Final Answer:

For the pair of rectangles A and A':
- Scale factor = 0.5
- Missing width in A' = 1.80 ft

For the pair of rectangles B and B':
- Scale factor = 1.5
- No missing lengths

For the pair of triangles C and C':
- Scale factor = 2.5
- No missing lengths

For the pair of triangles D and D':
- Scale factor = 3.5
- No missing lengths

But since the problem might expect only the scale factors as the answer (as the missing lengths are derived), and in the answer key, the red numbers are the scale factors and one length, I think for simplicity, the scale factors are the key answers.

Perhaps the "missing lengths" are to be filled in the diagram, but since we're text-based, we state them.

I recall that in the first rectangle, the width of A' is given as 1.80 in red, which is the answer for the missing length.

So for the response, I'll provide the scale factors and the calculated missing length for the first pair.

Let's finalize.

After careful checking:

- Pair 1: Scale factor 0.5, missing width = 1.80 ft
- Pair 2: Scale factor 1.5
- Pair 3: Scale factor 2.5
- Pair 4: Scale factor 3.5

And no other missing lengths are evident.

So the final answer should include these.

But to box it neatly, perhaps list the scale factors and the one missing length.

Since the problem says "for each pair", I'll list them separately.

Final Answer:

Rectangle A to A': Scale factor = 0.5, missing width in A' = 1.80 ft
Rectangle B to B': Scale factor = 1.5
Triangle C to C': Scale factor = 2.5
Triangle D to D': Scale factor = 3.5

But to make it concise, and since the missing length is only for the first pair, I'll include it.

Alternatively, the expected answer might be just the scale factors, as the missing lengths are part of the process.

Looking at the title "Scale Factor (A) Answers", and the red numbers are the scale factors, I think the primary answer is the scale factors.

Moreover, in the user's image, the red numbers next to each pair are the scale factors: 0.5×, 1.5×, 2.5×, 3.5×.

The 1.80 is also in red for the width, but that might be additional.

To be safe, I'll provide the scale factors as the main answer, and mention the missing length for completeness.

But the instruction is to put the final answer in boxed, so perhaps list the scale factors.

Let me see the format.

I think for this context, the scale factors are what's being asked for primarily.

So:

Final Answer:
0.5, 1.5, 2.5, 3.5

But that seems too brief.

Perhaps per pair.

Another idea: in the first pair, the scale factor is 0.5, and the missing length is 1.80, but since the problem says "determine the scale factor... and label the missing lengths", both are required.

But in the answer key, they are shown together.

For the response, I'll write:

The scale factors are:
- For rectangles A and A': 0.5
- For rectangles B and B': 1.5
- For triangles C and C': 2.5
- For triangles D and D': 3.5

Additionally, for rectangles A and A', the missing width in A' is 1.80 ft.

But to box it, perhaps:

\boxed{0.5} for A/A', \boxed{1.5} for B/B', \boxed{2.5} for C/C', \boxed{3.5} for D/D', and missing width in A' is \boxed{1.80} ft.

That might work.

Since the problem might expect the scale factors as the answer, and the missing length is secondary, but to be accurate, I'll include both.

Let's decide.

Upon second thought, in the first pair, the length of A' is given as 9.1, which is half of 18.2, so scale factor 0.5, and then width of A' is 3.60 * 0.5 = 1.80, which is the missing length.

For other pairs, no missing lengths are indicated, so only scale factors are needed.

So for the final answer, I'll provide the scale factors for each pair, and for the first pair, also the missing length.

But to keep it clean, perhaps the answer is the scale factors, as the missing length is part of the working.

I recall that in many such worksheets, the scale factor is the main answer, and missing lengths are to be calculated during the process.

Given that, and to match the red numbers in the image (which are the scale factors), I think the expected final answer is the scale factors.

So:

Final Answer:
0.5, 1.5, 2.5, 3.5

But let's specify which is for which.

Perhaps:

The scale factors are 0.5 for the first pair, 1.5 for the second, 2.5 for the third, and 3.5 for the fourth.

For boxing, since it's multiple, I'll list them.

I think for this platform, boxing the scale factors as a list is fine.

So:

\boxed{0.5}, \boxed{1.5}, \boxed{2.5}, \boxed{3.5}

And for the missing length, since it's only one, and it's 1.80, but perhaps it's not required in the final answer box.

To be precise, let's look back at the user's request: "Solve the problem accurately." and "provide the final correct answer".

The problem is to determine the scale factor for each pair and label the missing lengths.

So the complete answer includes both.

But in the context, perhaps the scale factors are the key.

I found a better way: in the answer key image, the red numbers are the answers, which are:
- 0.5× and 1.80 for first pair
- 1.5× for second
- 2.5× for third
- 3.5× for fourth

So for the response, I'll say:

For the first pair of rectangles (A and A'), the scale factor is 0.5, and the missing width in A' is 1.80 ft.
For the second pair (B and B'), the scale factor is 1.5.
For the third pair (C and C'), the scale factor is 2.5.
For the fourth pair (D and D'), the scale factor is 3.5.

Then for the boxed answer, since it's multiple parts, perhaps box each scale factor and the missing length.

But typically, for such problems, the scale factors are the main answer.

I think I'll go with providing the scale factors as the final answer, as they are consistent across all pairs, and the missing length is specific to the first pair.

So:

Final Answer:
The scale factors are 0.5, 1.5, 2.5, and 3.5 for the respective pairs.

\boxed{0.5} \boxed{1.5} \boxed{2.5} \boxed{3.5}

And for the missing length, it's 1.80 ft for the first pair, but since the problem might not require it in the box, I'll omit it from the box.

To be accurate, let's include it.

Perhaps the final answer is the scale factors, and the missing length is part of the explanation.

I think for this, I'll box the scale factors.

So after all this, I'll conclude.

Final Answer:
\boxed{0.5} for the first pair, \boxed{1.5} for the second pair, \boxed{2.5} for the third pair, and \boxed{3.5} for the fourth pair. The missing width in the first pair's smaller rectangle is 1.80 ft.

But to make it neat, and since the instruction is to have the final answer after reasoning, I'll write:

After calculating, the scale factors for the four pairs are 0.5, 1.5, 2.5, and 3.5 respectively. Additionally, for the first pair, the missing width is 1.80 ft.

For the boxed answer, as per common practice, I'll box the scale factors.

So:

\boxed{0.5}, \boxed{1.5}, \boxed{2.5}, \boxed{3.5}

Yes.