Scale factor quiz with two problems involving similar shapes and their dimensions.

Let's solve each problem step by step to find the scale factor between the two similar shapes. The scale factor is the ratio of the corresponding sides of two similar figures.

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Problem 1:

We are given two triangles:

- Triangle 1: sides = 10, 10, 5
- Triangle 2: sides = 50, 50, 25

These are similar triangles (same shape), and we want to find the scale factor from the smaller triangle to the larger one.

#### Step 1: Compare corresponding sides
Let’s take the corresponding sides:

- 10 → 50
- 10 → 50
- 5 → 25

Now calculate the ratio:
$$
\frac{50}{10} = 5 \quad \text{and} \quad \frac{25}{5} = 5
$$

So, each side of the larger triangle is 5 times the corresponding side of the smaller triangle.

Therefore, the scale factor from the small triangle to the large triangle is 5:1, or written as a ratio 1:5 if going from large to small? Wait — let’s clarify.

But the question says: "What is the scale factor?" — usually this means from the first figure to the second unless specified otherwise.

In this case:
- First triangle (smaller) → Second triangle (larger)
- So scale factor = $ \frac{\text{large}}{\text{small}} = \frac{50}{10} = 5 $

So the scale factor is 5:1, but that’s not an option.

Wait — look at the options:

> A) 1:4
> B) 1:6
> C) 1:5
> D) 10:25

Hmm. But 10:25 simplifies to 2:5, which is not 5:1.

But wait — maybe they're asking for the ratio of the smaller to the larger?

Let’s check:
If we go from large to small, then:
$$
\frac{10}{50} = \frac{1}{5} \Rightarrow \text{Scale factor } 1:5
$$

And option C is 1:5

So likely, the question is asking for the scale factor from the larger triangle to the smaller one, or it's just asking for the simplified ratio of corresponding sides in order.

But actually, scale factor is typically expressed as new : original.

But here, since both triangles are shown, and the first is smaller, and second is larger, and the answer choices include 1:5, which is less than 1, that suggests they want smaller to larger?

Wait — no. If the scale factor is 1:5, that means the second triangle is 5 times bigger than the first.

So if the first triangle has side 10, and second has 50, then the scale factor from first to second is 5:1.

But 5:1 is not among the choices.

But 1:5 is.

So perhaps the order is reversed?

Wait — maybe they're asking for the ratio of the smaller to the larger, so:

$$
\frac{10}{50} = \frac{1}{5} \Rightarrow 1:5
$$

Yes! That matches Option C.

So scale factor from the larger triangle to the smaller one is 1:5.

But usually, scale factor is from original to image.

But in this context, since both are shown, and the first is smaller, and second is larger, the scale factor from first to second should be 5:1.

But that’s not an option.

Wait — Option D is 10:25, which simplifies to 2:5, which is not correct.

Wait — maybe the scale factor is being asked as smaller to larger, so 1:5.

But 1:5 would mean the second triangle is 5 times the first — yes, that’s true.

But 1:5 means "for every 1 unit on the small triangle, there are 5 units on the large triangle".

So scale factor = 1:5 — but only if you’re going from small to large.

But typically, we write scale factor as image / preimage.

Here, if the first triangle is the preimage, and the second is the image, then:
$$
\text{Scale factor} = \frac{50}{10} = 5 \Rightarrow 5:1
$$

But 5:1 is not an option.

But 1:5 is.

Wait — unless the question is asking for the ratio of the smaller to the larger, which is 1:5.

Alternatively, perhaps the quiz is asking for the reduction scale factor.

But looking at the options, C) 1:5 is the only one that makes sense numerically.

Let’s double-check:

- Side 10 → 50 → ratio = 50/10 = 5 → so scale factor is 5
- But 1:5 is the reciprocal.

Wait — maybe the scale factor is defined as smaller : larger, so 1:5

But that’s not standard.

Alternatively, perhaps the answer is C: 1:5, meaning the second triangle is scaled up by a factor of 5, so the ratio of original to image is 1:5.

Wait — confusion arises.

Let’s rephrase:

If the small triangle is the original, and the large triangle is the scaled image, then:

$$
\text{Scale factor} = \frac{\text{image}}{\text{original}} = \frac{50}{10} = 5
$$

So the scale factor is 5, or 5:1.

But that’s not an option.

But 1:5 is.

Wait — what if the order is reversed? Maybe the second triangle is the original, and the first is the image?

Then scale factor = $ \frac{10}{50} = \frac{1}{5} = 1:5 $

That would make sense.

But the way the problem is laid out: first triangle is small, second is large — and it asks "what is the scale factor?"

But without specifying direction, we assume from left to right.

But still, 5:1 isn't an option.

Wait — look at Option D: 10:25

That’s 10:25, which is 2:5, not 1:5.

But 10:25 = 2:5 ≈ 0.4, while 1:5 = 0.2.

Not matching.

Wait — maybe the scale factor is the ratio of corresponding sides, and they want it simplified.

But 10:50 = 1:5 → so 1:5

Ah! So even though the side is 10 to 50, the ratio is 1:5.

So the scale factor is 1:5 — but again, this implies small to large?

No — 1:5 means smaller to larger.

But if the first triangle is the original, then scale factor is 5:1.

But 5:1 is not an option.

Unless the question is asking for the ratio of the smaller to the larger, which is 1:5, and that's acceptable.

But let’s see the next problem.

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Problem 2:

Two quadrilaterals:

- First: sides 8, 28, 18, 30
- Second: sides 12, 42, 27, 45

We need to find the scale factor.

Let’s compare corresponding sides:

- 8 → 12 → ratio = $ \frac{12}{8} = 1.5 = \frac{3}{2} $
- 28 → 42 → $ \frac{42}{28} = \frac{3}{2} $
- 18 → 27 → $ \frac{27}{18} = \frac{3}{2} $
- 30 → 45 → $ \frac{45}{30} = \frac{3}{2} $

So all ratios are $ \frac{3}{2} $, so scale factor is 3:2

But look at the options:

> A) 1:4
> B) 2:3
> C) 1:5
> D) 3:4

None of them is 3:2.

Wait — but B is 2:3, which is reciprocal.

So if the scale factor is from the first to the second, it’s 3:2

But 3:2 is not an option.

But 2:3 is.

So again, same issue.

Wait — unless the scale factor is from the larger to the smaller?

But the second quadrilateral is larger, so from small to large is 3:2, from large to small is 2:3

So if the question is asking for the scale factor from the first to the second, it should be 3:2, not listed.

But 2:3 is listed.

So maybe the question is asking for the ratio of the smaller to the larger, i.e., 2:3?

But that would be incorrect because 8 → 12 is 1.5x, not 2/3.

Wait — 8 × (2/3) = 5.33… not 12.

So 2:3 is not correct.

Wait — let’s try 3:4?

3:4 = 0.75 — 8 × 0.75 = 6 ≠ 12.

No.

Wait — let’s think.

Is there a possibility I misread the sides?

First quadrilateral:
- Left side: 8
- Top: 28
- Right: 18
- Bottom: 30

Second:
- Left: 12
- Top: 42
- Right: 27
- Bottom: 45

Now check ratios:

- 8 → 12: $ \frac{12}{8} = 1.5 = \frac{3}{2} $
- 28 → 42: $ \frac{42}{28} = 1.5 $
- 18 → 27: $ \frac{27}{18} = 1.5 $
- 30 → 45: $ \frac{45}{30} = 1.5 $

So scale factor from first to second is 3:2

But 3:2 is not an option.

Options are:

A) 1:4
B) 2:3
C) 1:5
D) 3:4

None is 3:2.

But 2:3 is close — but it’s the reciprocal.

Wait — could the scale factor be defined as original to image, but the image is the first, and original is the second?

But the diagram shows the first as smaller, second as larger.

So likely, first is original, second is image, so scale factor = 3:2

But not in options.

Wait — maybe the scale factor is simplified as a fraction?

But 3:2 is already simplified.

Wait — perhaps the question is asking for the scale factor from the larger to the smaller?

Then:

$ \frac{8}{12} = \frac{2}{3} \Rightarrow 2:3 $

Which is Option B

So if the scale factor is from the second (larger) to the first (smaller), then it's 2:3

Similarly, in Problem 1:

- 10 → 50: ratio = 5 → so from small to large is 5:1
- But if from large to small: 10/50 = 1/5 → 1:5 → Option C

So in both cases, the scale factor is being asked as from the larger to the smaller, or the reduction factor.

But that’s not standard.

Alternatively, maybe the quiz assumes the scale factor is the ratio of the smaller to the larger, which is less than 1, and expressed as a ratio.

So for Problem 1:
- Small to large: 10:50 = 1:5 → Option C

For Problem 2:
- Small to large: 8:12 = 2:3 → Option B

Ah! That makes sense!

So the scale factor is defined as the ratio of the smaller figure to the larger figure — i.e., small : large

Even though that’s not the standard definition, in some contexts, especially multiple-choice quizzes, they might define it that way.

Let’s verify:

Problem 1:

- Smaller triangle side: 10
- Larger triangle side: 50
- Ratio: $ \frac{10}{50} = \frac{1}{5} = 1:5 $ → Option C

Problem 2:

- Smaller quadrilateral side: 8
- Larger: 12
- Ratio: $ \frac{8}{12} = \frac{2}{3} $ → Option B

But wait — 8:12 = 2:3, yes.

But is that the scale factor? Only if we're going from small to large, but 2:3 means 2/3, which is less than 1, so it would mean the image is smaller, but it's not.

Wait — no: if the scale factor is 2:3, that means for every 2 units on the small, there are 3 on the large — so the large is 1.5 times the small.

So scale factor = 3/2 from small to large.

But 2:3 is small to large? No — 2:3 means small : large = 2:3, so large is 1.5× small.

So yes, 2:3 can represent the ratio of corresponding sides, with small : large.

But in terms of scale factor, it’s usually image / preimage.

But if the preimage is the small, and image is the large, then scale factor = 3/2.

But if the scale factor is defined as small : large, then it’s 2:3.

But that’s confusing.

Alternatively, perhaps the scale factor is the ratio of corresponding lengths, and they want it simplified, and in lowest terms, but without specifying direction.

But in Problem 1, 10:50 = 1:5 → C

In Problem 2, 8:12 = 2:3 → B

And both match.

But in Problem 2, the ratio of small to large is 8:12 = 2:3, which is Option B

Similarly, in Problem 1, 10:50 = 1:5 → Option C

So likely, the scale factor is being interpreted as the ratio of the smaller figure to the larger one, or simply the ratio of corresponding sides in simplest form, and they expect 1:5 and 2:3.

But wait — in Problem 1, the sides are 10 and 50, so ratio is 1:5 — C

In Problem 2, the corresponding sides are 8 and 12, so ratio is 8:12 = 2:3 — B

So answers are:

1. C) 1:5
2. B) 2:3

But let’s confirm if all sides support this.

Problem 2:

- 8 → 12 → 8/12 = 2/3
- 28 → 42 → 28/42 = 2/3
- 18 → 27 → 18/27 = 2/3
- 30 → 45 → 30/45 = 2/3

Yes, all ratios are 2:3 when small : large

So the scale factor from small to large is 3:2, but the ratio of small to large is 2:3

And since the question says "What is the scale factor?", and gives options like 2:3, it's likely they want the ratio of corresponding sides in simplest form, and they consider the smaller to larger.

But technically, scale factor is image / preimage.

But in many educational contexts, especially with multiple choice, they may ask for the ratio of the two figures, and accept 2:3 as the simplified ratio.

Alternatively, maybe the scale factor is defined as how much the figure is scaled, so if the small is the original, scale factor is 3/2, but that’s not an option.

Given the options, the only way to match is:

Final Answers:

1. C) 1:5
Because 10:50 = 1:5

2. B) 2:3
Because 8:12 = 2:3, and all other sides give the same ratio

So the scale factor is taken as the ratio of corresponding sides in simplest form, and from the smaller to the larger.

Thus:

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✔ Answer Key:

1. C) 1:5
2. B) 2:3

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🔍 Explanation:

- Scale factor is the ratio of corresponding lengths of two similar figures.
- For Problem 1:
- Corresponding sides: 10 and 50 → $ \frac{10}{50} = \frac{1}{5} $ → 1:5
- For Problem 2:
- Corresponding sides: 8 and 12 → $ \frac{8}{12} = \frac{2}{3} $ → 2:3

Even though the actual scaling from small to large is 3:2, the ratio of the smaller to the larger is 1:5 and 2:3, respectively, which matches the options.

So the intended answers are:

> 1. C) 1:5
> 2. B) 2:3