Answer :
Final answer:
The area of triangle AFD, which is a right triangle with a 30-degree angle and a hypotenuse of length 14, is calculated using the properties of a 30-60-90 triangle to be 49\sqrt{3}. The correct option is d.
Explanation:
The area of triangle AFD can be found using the formula for the area of a right triangle, which is \(\frac{1}{2} \times base \times height\). In triangle AFD, we're given that \(\angle F = 90\) degrees and \(m\angle D = 30\) degrees. Since AD is the hypotenuse of the right triangle and its measure is 14, we can use the properties of a 30-60-90 right triangle to determine the lengths of the other two sides.
In a 30-60-90 triangle, the length of the side opposite the 30-degree angle is half of the hypotenuse, and the length of the side opposite the 60-degree angle (the height in this case) is \(\sqrt{3}/2\) times the hypotenuse. Therefore, the area of triangle AFD is 49√3 The correct answer is option (b). The calculation involves using the trigonometric ratio tangent to find the height of the triangle and then applying the area formula for a triangle with the given base and height.
