Answer:
Step-by-step explanation:
To find the perimeter of the shaded region, we need to calculate the lengths of the sides of the triangle and the arc.
Given:
AB = BC = 6 cm
Angle ABC = 1.8 radians
Since AB = BC, triangle ABC is an isosceles triangle. Therefore, angle BAC = angle BCA.
The sum of the angles in a triangle is 180 degrees or π radians. So, we can find angle BAC as follows:
180 degrees = angle BAC + angle ABC + angle BCA
180 degrees = angle BAC + 1.8 radians + angle BAC
180 degrees - 1.8 radians = 2 * angle BAC
angle BAC = (180 degrees - 1.8 radians) / 2
Now, we can find the length of side AC using the cosine rule:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(angle ABC)
AC^2 = 6^2 + 6^2 - 2 * 6 * 6 * cos(1.8 radians)
AC^2 = 72 - 72 * cos(1.8 radians)
AC = √(72 - 72 * cos(1.8 radians))
The perimeter of the shaded region is equal to the sum of the lengths of sides AB, BC, and the arc CD.
Perimeter = AB + BC + arc CD
To find the length of the arc CD, we need to find the circumference of the circle with radius AC and multiply it by the ratio of the angle ABC to the total angle of a circle (2π radians or 360 degrees).
Circumference of the circle = 2 * π * AC
Arc CD = (angle ABC / 2π) * Circumference of the circle
Arc CD = (1.8 radians / 2π) * (2 * π * AC)
Arc CD = 1.8 radians * AC
Now, we can calculate the perimeter of the shaded region:
Perimeter = AB + BC + arc CD
Perimeter = 6 cm + 6 cm + 1.8 radians * AC
Substituting the value of AC we found earlier:
Perimeter = 6 cm + 6 cm + 1.8 radians * √(72 - 72 * cos(1.8 radians))
Please note that the final answer will be in centimeters, as we're adding lengths in centimeters.
