Similar Triangles and Figures, Enlargement Ratios & Proportions ... - Free Printable

The image presents a problem involving similar triangles. Let's solve it step by step and explain the reasoning.

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🔷 Given:

Two triangles:
- Triangle $ \triangle ABC $ with side lengths:
- $ AB = 3 $
- $ BC = 6 $
- $ AC = 4 $

- Triangle $ \triangle DEF $ with side lengths:
- $ DE = 9 $
- $ EF = y $ (unknown)
- $ DF = x $ (unknown)

Also given:
- $ \triangle ABC \sim \triangle DEF $ → The triangles are similar.
- Corresponding angles are equal (indicated by matching arc marks).
- The proportionality of sides is given as:
$$
\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}
$$

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🔷 Goal:

Find the values of $ x $ and $ y $, the missing side lengths in $ \triangle DEF $.

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🔷 Step 1: Use the similarity ratio

Since $ \triangle ABC \sim \triangle DEF $, their corresponding sides are proportional.

Let’s find the scale factor from $ \triangle ABC $ to $ \triangle DEF $ using the known pair:

$$
\frac{DE}{AB} = \frac{9}{3} = 3
$$

So, the scale factor is 3. This means each side of $ \triangle DEF $ is 3 times the corresponding side in $ \triangle ABC $.

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🔷 Step 2: Find $ y = EF $

$ EF $ corresponds to $ BC $

$$
EF = 3 \times BC = 3 \times 6 = 18
$$

So, $ y = 18 $

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🔷 Step 3: Find $ x = DF $

$ DF $ corresponds to $ AC $

$$
DF = 3 \times AC = 3 \times 4 = 12
$$

So, $ x = 12 $

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✔ Final Answer:

- $ x = 12 $
- $ y = 18 $

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🔷 Check Proportionality:

Verify all ratios are equal:

$$
\frac{DE}{AB} = \frac{9}{3} = 3 \\
\frac{EF}{BC} = \frac{18}{6} = 3 \\
\frac{DF}{AC} = \frac{12}{4} = 3
$$

✔ All ratios are equal → confirms similarity and correctness.

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🟩 Summary:

Using the fact that the triangles are similar and the scale factor is 3, we multiplied each side of $ \triangle ABC $ by 3 to get the corresponding sides in $ \triangle DEF $.

Answer:
- $ x = 12 $
- $ y = 18 $