To determine the exact length of each base edge of a right triangular pyramid where the base is an equilateral triangle, given a lateral area of [tex]\(180 \text{ cm}^2\)[/tex] and a slant height of [tex]\(10 \text{ cm}\)[/tex], follow these steps:
1. Understand the formula for the lateral area of a right triangular pyramid:
[tex]\[
\text{Lateral Area} = \frac{\text{Base Perimeter} \times \text{Slant Height}}{2}
\][/tex]
2. Identify the given values:
[tex]\[
\text{Lateral Area} = 180 \text{ cm}^2
\][/tex]
[tex]\[
\text{Slant Height} = 10 \text{ cm}
\][/tex]
3. Determine the formula for the perimeter of the base:
Since the base is an equilateral triangle, and let [tex]\( a \)[/tex] be the length of each base edge:
[tex]\[
\text{Base Perimeter} = 3a
\][/tex]
4. Substitute the known values into the lateral area formula:
[tex]\[
180 = \frac{(3a) \times 10}{2}
\][/tex]
5. Simplify the equation:
[tex]\[
180 = \frac{30a}{2}
\][/tex]
[tex]\[
180 = 15a
\][/tex]
6. Solve for [tex]\( a \)[/tex] (the length of each base edge):
[tex]\[
a = \frac{180}{15}
\][/tex]
[tex]\[
a = 12 \text{ cm}
\][/tex]
Thus, the exact length of each base edge of the pyramid is [tex]\(12 \text{ cm}\)[/tex].
