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The problem asks for the height of a smaller triangle given that the ratio of the areas of two triangles with equal bases is 8:5, and the height of the larger triangle is 12 cm.

We can use the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Since the bases of the triangles are equal, the ratio of their areas depends only on the heights of the triangles. Let the height of the smaller triangle be \( h \).

The ratio of the areas of the triangles is:

\[
\frac{\text{Area of larger triangle}}{\text{Area of smaller triangle}} = \frac{8}{5}
\]

This simplifies to the ratio of the heights:

\[
\frac{12}{h} = \frac{8}{5}
\]

Now, solve for \( h \):

\[
12 \times 5 = 8 \times h
\]

\[
60 = 8h
\]

\[
h = \frac{60}{8} = 7.5 \, \text{cm}
\]

Thus, the height of the smaller triangle is 7.5 cm.

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Would you like further details or have any questions? Here are five related questions to expand on this problem:

1. How would the ratio change if the bases of the triangles were different?
2. Can the ratio of the heights be directly used if both base and height change?
3. How would you solve this problem if the ratio of areas was given for a trapezoid instead of a triangle?
4. If the height of the smaller triangle was known, how could you calculate the height of the larger triangle?
5. How does the formula for the area of a triangle help in comparing different properties of triangles?

**Tip:** When the bases are equal, comparing areas is the same as comparing heights. This makes calculations much simpler!

Ratio of Areas of two triangles of equal bases are in the ratio 8:5. Height of the larger triangle is 12 cm. Find the corresponding height of the smaller triangle.

This problem involves finding the height of a smaller triangle when the ratio of areas of two triangles with equal bases is given as 8:5, and the height of the larger triangle is 12 cm. Using proportions and the area formula for triangles, we solve the problem step-by-step.

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