Answer:
Length EA = z = 12
Explanation:
We have the right triangle ΔMEN which has side MN as hypotenuse and side ME = 6 as one of the legs
Therefore the other leg NE can be computed by the Pythagorean theorem as
NE ² = MN² - ME²
= (6√3)² - 6²
= 6²(√3)² - 6²
(√3)² = 3
Therefore
NE² = 6²(3) - 6² = 6²(3 - 1) = 6² (2) = 36 (2) = 72
ΔNEA is also a right triangle with side AN as hypotenuse and sides AE and NE as the legs
AN² = NE² + AE² = 72 + z²
AN² = z² + 72 (1)
Considering right triangle ΔMNA,
hypotenuse = MA
legs are MN and AN
MA² = MN² + AN²
AN² = MA² - MN²
MA = ME + EA = 6 + z
MN = 6√3
AN² = (6 + z)² - (6√3)²
(6 + z)² = 36 + 12z + z²
(6√3)² = 36 x 3 = 108
Hence
AN² = 36 + 12z + z² - 108
AN² = z² + 12z - 72 (2)
Since the left side of equations (1) and (2) are the same we can equate the right sides to each other:
z² + 72 = z² + 12z - 72
z² cancels out (opposite sides with same coefficient and sigh)
==> 72 = 12z -72
==> 144 = 12z (by adding 72 on both sides)
==> 12z = 144 (switching sides)
==> z = 144/12 = 12 (dividing both sides by 12)
Therefore z = 12 = length EA