We are given two triangles, and we know the following details:

1. First triangle:

   • Area = 10 square meters
   • Base = 4 meters
   • Height = ?

2. Second triangle:

   • Area = 90 square meters
   • Base = (assumed similar, so we assume it scales with the height)

To find the height of the larger triangle, we can use the relationship between the area of a triangle and its base and height. The formula for the area of a triangle is:

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

Step 1: Find the height of the smaller triangle

We know the area and the base of the smaller triangle. So, solving for height:

$10 = \frac{1}{2} \times 4 \times h_1$ $h_1 = \frac{10 \times 2}{4} = 5 \, \text{meters}$

Step 2: Use the ratio of areas to find the height of the larger triangle

Since the triangles are similar, the ratio of the areas is equal to the square of the ratio of the corresponding sides (including height). The ratio of the areas is:

$\frac{\text{Area of larger triangle}}{\text{Area of smaller triangle}} = \frac{90}{10} = 9$

Therefore, the ratio of the heights is the square root of this:

$\frac{h_2}{h_1} = \sqrt{9} = 3$

So, the height of the larger triangle is:

$h_2 = 3 \times h_1 = 3 \times 5 = 15 \, \text{meters}$

Thus, the height of the larger triangle is 15 meters.

Would you like further details or have any questions?

Here are some follow-up questions:

1. What is the formula for the area of a triangle?
2. How do the areas of similar triangles relate to their sides?
3. What is the height of the smaller triangle in this problem?
4. How did we use the ratio of areas to find the height of the larger triangle?
5. Can this method be used for non-similar triangles?

Tip: For similar geometric shapes, corresponding lengths (such as heights) scale by the same factor as the square root of the ratio of their areas.