To solve question 1:

The problem states that $\triangle ABC \sim \triangle DEF$, meaning the triangles are similar. For similar triangles, the ratios of corresponding sides are equal.

Corresponding side pairs:

• $AB : DE$
• $AC : DF$
• $BC : EF$

Given:

• $AB = 7 \, \text{cm}$
• $AC = 6 \, \text{cm}$
• $BC = 10 \, \text{cm}$
• $DE = 4 \, \text{cm}$ (one corresponding side provided)

We need to find $EF$ and $DF$.

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Step 1: Ratio of similarity

Using $AB : DE$, the ratio is: $\frac{AB}{DE} = \frac{7}{4}.$

Thus, the ratio of similarity is $\frac{7}{4}$. This same ratio applies to the other pairs of corresponding sides.

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Step 2: Find $EF$

Using $BC : EF$, we solve for $EF$: $\frac{BC}{EF} = \frac{7}{4}.$

Substitute $BC = 10$: $\frac{10}{EF} = \frac{7}{4}.$

Cross-multiply: $7 \cdot EF = 4 \cdot 10.$

$7 \cdot EF = 40 \implies EF = \frac{40}{7} \approx 5.7 \, \text{cm}.$

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Step 3: Find $DF$

Using $AC : DF$, we solve for $DF$: $\frac{AC}{DF} = \frac{7}{4}.$

Substitute $AC = 6$: $\frac{6}{DF} = \frac{7}{4}.$

Cross-multiply: $7 \cdot DF = 4 \cdot 6.$

$7 \cdot DF = 24 \implies DF = \frac{24}{7} \approx 3.4 \, \text{cm}.$

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Final Answer:

• $EF = 5.7 \, \text{cm}$
• $DF = 3.4 \, \text{cm}$

Would you like a detailed breakdown or help with another question?

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Related Questions:

1. How are similar triangles defined, and what properties do they share?
2. How do you find the ratio of similarity between two triangles?
3. What other geometric methods can be used to verify similarity?
4. How would the problem change if $DE$ was not provided?
5. What are some real-life applications of similar triangles?

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Tip:

When working with similar triangles, always align the corresponding sides properly to avoid errors in ratio calculation!