Final answer:
To find angle FAH in the cuboid, we use the Pythagorean theorem to calculate the length FH, then trigonometry to find the angle, which is approximately 47° to the nearest degree.
Step-by-step explanation:
Finding Angle FAH in a Cuboid
To find the size of angle FAH in a cuboid ABCDEFGH, we need to consider the three-dimensional geometry of the figure. Given that AD = 9cm and FD = 13cm, and knowing the angle GHF = 49°, we can deduce that triangles FDH and FAH are right-angled triangles because they are formed by the edges of the cuboid which meet at right angles.
First, we find the length of AH using the Pythagorean theorem. Since FD is the hypotenuse of the right-angled triangle FDH, we have:
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- FD² = DH² + FH²
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- 13² = 9² + FH²
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- 169 = 81 + FH²
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- FH² = 169 - 81
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- FH = √(169 - 81)
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- FH = √88 ≈ 9.38 cm
Triangle FAH shares the side AH with triangle FDH and also has a right angle at A, so we can calculate the angle FAH using trigonometry:
cos(FAH) = AD / FD
cos(FAH) = 9 / 13
FAH = cos⁻¹(9 / 13)
FAH ≈ cos⁻¹(0.6923)
FAH ≈ 46.57°
Therefore, the size of angle FAH is approximately 47° when rounded to the nearest degree.