Let's solve this step by step.

Step 1: Analyze the triangle

The triangle $\triangle EGF$ is isosceles, with $EG = EF = 7 \, \text{cm}$. The altitude $EH$ divides the base $FG$ into two equal segments, so: $FH = HG = \frac{FG}{2}.$

We aim to calculate $FG$, the length of the base.

Step 2: Using Pythagoras' Theorem

In the right triangle $\triangle EHF$, the hypotenuse is $EF = 7 \, \text{cm}$, one leg is $EH = 4 \, \text{cm}$, and the other leg is $FH$.

Using Pythagoras' Theorem: $EF^2 = EH^2 + FH^2.$ Substitute the known values: $7^2 = 4^2 + FH^2.$ Simplify: $49 = 16 + FH^2.$ Solve for $FH^2$: $FH^2 = 49 - 16 = 33.$ Take the square root: $FH = \sqrt{33} \approx 5.7 \, \text{cm} \, (\text{to 1 decimal place}).$

Step 3: Calculate $FG$

Since $FH = HG$ (the altitude divides $FG$ into two equal parts): $FG = 2 \times FH = 2 \times 5.7 = 11.4 \, \text{cm}.$

Final Answer:

The length of $FG$ is: $\boxed{11.4 \, \text{cm}}$

Would you like further details or additional clarifications? Here are some related questions to explore:

1. How does the isosceles property help simplify calculations in this problem?
2. Can we verify the answer by calculating the total perimeter of the triangle?
3. How would the answer change if $EH$ were not given as perpendicular?
4. What other triangle properties can be derived using Pythagoras' theorem?
5. How would you solve this problem if only the area of the triangle were given?

Tip:

For any triangle problem involving altitudes and equal sides, split the triangle into smaller right triangles to simplify your calculations!