Questions: Question number 5. Right triangle ABC is given below. If AB = 4, find the area of this triangle. None of the above O 8√3 O 4√3 O 2√3 O √3 O 3 A 60° B C

Solution Steps

In the right triangle ABC, angle C is 90 degrees, and angle A is 60 degrees. Therefore, angle B is 30 degrees. We are given that AB = 4. We can use the trigonometric ratio for the sine function in a 30-60-90 triangle. \\( \sin(30^{\circ}) = \frac{AC}{AB} \\) \\( \frac{1}{2} = \frac{AC}{4} \\) \\( AC = 4 \times \frac{1}{2} = 2 \\)

We can use the trigonometric ratio for the cosine function in a 30-60-90 triangle. \\( \cos(30^{\circ}) = \frac{BC}{AB} \\) \\( \frac{\sqrt{3}}{2} = \frac{BC}{4} \\) \\( BC = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \\)

The area of triangle ABC is given by: Area = \\( \frac{1}{2} \times \text{base} \times \text{height} \\) In our case, the base is BC, and the height is AC. Area = \\( \frac{1}{2} \times BC \times AC \\) Area = \\( \frac{1}{2} \times 2\sqrt{3} \times 2 \\) Area = \\( 2\sqrt{3} \\)

Final Answer