Final answer:
Using the Pythagorean theorem, we find the other leg of the right triangle and then calculate the tangent of the angle as the square root of 5 divided by 2. However, this value does not match any of the options provided, indicating there may be an error in the question or a need for additional rationalization.
Step-by-step explanation:
The student is asked to find the tangent of an angle in a right triangle with a leg of length 12 and a hypotenuse of length 18. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
First, we need to use the Pythagorean theorem to find the unknown leg. The theorem states that in a right triangle, the sum of the squares of the legs (a² + b²) is equal to the square of the hypotenuse (c²). Considering the leg opposite the angle as 'a', the leg adjacent as 'b', and the hypotenuse as 'c', and knowing that c = 18:
a² + b² = c²
a² + 12² = 18²
a² = 18² - 12²
a² = 324 - 144
a² = 180
a = √180
Now, we calculate the tangent of the angle:
tan(Θ) = opposite / adjacent
tan(Θ) = √180 / 12
After simplifying, we find that:
tan(Θ) = √180 / 12
tan(Θ) = √(9×20) / 12
tan(Θ) = 3√(20) / 12
tan(Θ) = 3√4√5 / 12
tan(Θ) = 3×2√5 / 12
tan(Θ) = 6√5 / 12
tan(Θ) = √5 / 2
tan(Θ) = √5 / (√4)
tan(Θ) = √5 / 2
The exact value of √5 is not one of the options provided, which suggests the answer requires rationalization or approximation. Without more information on the rationalization or the context in which this question is being asked, we cannot definitively match the calculated tangent to one of the options provided.