8. A right triangular pyramid has a base that is an equilateral triangle. The pyramid has a lateral area of [tex]180 \, \text{cm}^2[/tex] and a slant height of [tex]10 \, \text{cm}[/tex]. What is the exact length of each base edge of this pyramid? Include units in your answer. (3 pts)



Answer :

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].