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To find the height \( h \) of the figure, we need to use the information given:

1. The three areas \( D \) (triangle), \( Q \) (square), and \( R \) (rectangle) are all equal.
2. The base \( a \) of the figure is 30 cm.

Let's denote the area of each region as \( A \).

Since the areas are equal, we can express them as follows:
\[ A_D = A_Q = A_R \]

For the square \( Q \):
\[ A_Q = s^2 \]
where \( s \) is the side length of the square.

For the rectangle \( R \):
\[ A_R = s \times h \]

For the triangle \( D \):
\[ A_D = \frac{1}{2} s \times h \]

Since all three areas are equal, we have:
\[ s^2 = s \times h = \frac{1}{2} s \times h \]

From \( s \times h = \frac{1}{2} s \times h \), it is clear that there is a contradiction unless \( h = 2s \).

Thus, \( h = 2s \).

Next, we use the total base \( a \):
\[ a = s + s + s = 3s = 30 \, \text{cm} \]

So, the side length \( s \) is:
\[ s = \frac{30}{3} = 10 \, \text{cm} \]

Now, we substitute \( s \) back to find \( h \):
\[ h = 2s = 2 \times 10 = 20 \, \text{cm} \]

Therefore, the height \( h \) of the figure is \( 20 \, \text{cm} \).

Would you like more details or have any questions?

Here are five questions you might find interesting to ask:
1. How do you calculate the area of a triangle given the base and height?
2. Can you explain how to derive the formula for the area of a rectangle?
3. What are some other methods to solve for the height of geometric figures?
4. How would the calculations change if the base \( a \) was a different length?
5. What are some real-life applications of calculating areas and heights of geometric shapes?

**Tip:** Always check for any contradictions in the equations to verify the accuracy of your solution.

Learn how to calculate the height (h) of a geometric figure where the areas of a triangle, square, and rectangle are equal. This problem involves understanding and applying the formulas for area calculations in geometry. Suitable for high school students.

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