Final answer:
To find the area of an isosceles triangle with one angle measuring 150° and legs of length 16, use the Law of Cosines to find the base, and then trigonometry to find the height. Once the base and height are known, apply the area formula for a triangle.
Step-by-step explanation:
To calculate the area of an isosceles triangle with a leg length of 16 and an angle measure of 150°, we need to determine the base and height of the triangle. Since the triangle is isosceles, the two equal sides are the legs, and the base will be the third side. We can use the Law of Cosines for this purpose.
The Law of Cosines is given by:
c² = a² + b² - 2ab cos(y)
Here, we can consider our triangle to have sides a and b (the equal legs) and base c, with y being the angle opposite to base c.
So, we have:
c² = 16² + 16² - 2(16)(16) cos (150°)
The next step is to calculate the height (h) of the triangle using trigonometry, particularly since we have c and an angle. After calculating c and h, the area (A) can be found using the formula:
A = 1/2 × base × height
Once c and h have been determined, you multiply them and then divide by 2 to obtain the area of the triangle.