Given: MPNA pyramid All edges congruent AO = 36, AO ⊥ (MNP) Find: m∠AMO, AΔANM
1 answer:
Answer:
m∠AMO ≈ 54.7°
AΔANM = 486√3
Step-by-step explanation:
The edges are congruent, so all four faces are congruent equilateral triangles. We'll say the length of each edge is 2r.
The height of the pyramid h is given to be 36.
The perpendicular distance from O to line MP is called the apothem (a). Using 30-60-90 triangles, b = 2a and r = a√3.
Use cosine to find m∠AMO.
cos(∠AMO) = b / (2r)
cos(∠AMO) = (2a) / (2a√3)
cos(∠AMO) = 1 / √3
m∠AMO ≈ 54.7°
Use Pythagorean theorem to find the apothem.
(2r)² = b² + h²
(2a√3)² = (2a)² + 36²
12a² = 4a² + 1296
8a² = 1296
a² = 162
a = 9√2
So the edge length is:
2r = 2√3 (9√2)
2r = 18√6
The area of the equilateral triangle ΔANM is half the apothem times the perimeter:
A = ½aP
A = ½ (9√2) (3 × 18√6)
A = 243√12
A = 486√3
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