Let's analyze the problem step by step to calculate $EF$ and $AC$.

Given Data:

1. $AB = 15 \, \text{cm}$
2. $DC = 18 \, \text{cm}$
3. $EG = 5 \, \text{cm}$
4. $GC = 10 \, \text{cm}$
5. $AB \parallel EF \parallel CD$

Required:

1. $EF$
2. $AC$

Solution:

1. Calculate $EF$:

Since $AB \parallel EF \parallel CD$, the triangles $\triangle ABF$ and $\triangle DCF$ are similar by the Basic Proportionality Theorem (BPT).

Thus, the ratios of corresponding sides are equal: $\frac{AB}{EF} = \frac{EF}{CD}$

Let $EF = x$. Substituting the given values: $\frac{15}{x} = \frac{x}{18}$

Cross-multiply: $15 \cdot 18 = x^2$

Simplify: $x^2 = 270$

Take the square root: $x = \sqrt{270} = 3\sqrt{30} \, \text{cm} \, (\text{approx. } 16.43 \, \text{cm})$

Thus, $EF = 3\sqrt{30} \, \text{cm}$.

2. Calculate $AC$:

In $\triangle EGC$, $EG = 5 \, \text{cm}$ and $GC = 10 \, \text{cm}$. Since both segments are given, $AC = EG + GC$: $AC = 5 + 10 = 15 \, \text{cm}.$

Final Answers:

1. $EF = 3\sqrt{30} \, \text{cm} \, (\text{approx. } 16.43 \, \text{cm})$
2. $AC = 15 \, \text{cm}$

Would you like me to provide more detailed explanations for each step or clarify any part of the solution?

5 Related Questions:

1. How does the Basic Proportionality Theorem apply to similar triangles?
2. Why is it valid to assume triangles $\triangle ABF$ and $\triangle DCF$ are similar?
3. What are the properties of parallel lines that help in solving this problem?
4. How do you calculate the square root of a number like $270$ more accurately?
5. Can the same method be used to find $EF$ if other dimensions are changed?

Tip:

When dealing with similar triangles, always look for proportional sides and equal angles as key indicators to apply similarity rules effectively.