Short Answer 
The diameter of the semicircle is calculated using the distance formula, resulting in ‚ao65. The radius is half of the diameter, leading to an area of 8.125œA square units for the semicircle, while the area of the triangle formed by the endpoints is 14 square units, making the total area 14 + 8.125œA square units.
Step 1: Calculate the Diameter of the Semicircle
To determine the diameter of the semicircle, we must connect the two endpoints: (3, 2) and (-4, -2). We apply the distance formula to find the length between these two points. The formula is given by:
- Diameter = ‚ao{(-4 Р3)¬≤ + (-2 Р2)¬≤}
- Calculate: = ‚ao{(-7)¬≤ + (-4)¬≤}
- Result: = ‚Äöao{49 + 16} = ‚Äöao65
Step 2: Determine the Radius and Area of the Semicircle
The radius of the semicircle is half of the diameter, calculated as follows:
- Radius = ‚Äöao65 / 2
- Using the radius, we will find the area of the semicircle using the formula: Area = (1/2)œAr¬≤.
- Substituting the radius: Area = (1/2)œA(‚ao65 / 2)¬≤ = 8.125œA square units.
Step 3: Calculate Area of the Triangle and Total Area
Next, we need to determine the area of the triangle formed by the endpoints. The base is calculated as:
- Base = 3 – (-4) = 7
- Height = 2 – (-2) = 4
- Area of the triangle = (1/2) ‚àöo Base ‚àöo Height = (1/2) ‚àöo 7 ‚àöo 4 = 14 square units.
Finally, the total area is the sum of the area of the semicircle and the triangle: Total Area = 14 + 8.125œA square units.