Short Answer 
The diameter of the semicircle is calculated to be approximately 8.544, leading to a radius of about 4.272. The total area, combining both the triangle and the semicircle, is determined to be 12 + (9.125œA) square units.
Step 1: Calculate the Diameter
To find the diameter of the semicircle, start by determining the distance between the points (-1, -5) and (2, 3). This distance serves as the hypotenuse of the triangle formed. Utilize the formula for distance:
- Apply the Pythagorean theorem: d = ‚ao(base¬≤ + height¬≤), where base is 3 and height is 8.
- Calculate to find diameter: d ‚Äöaa 8.544.
Step 2: Determine the Radius
Once the diameter is established, the next step is to find the radius of the semicircle. The radius is simply half of the diameter calculated in the previous step. Use the following calculations:
- Formula for radius: r = d / 2.
- Substituting the diameter, r = 8.544 / 2 ‚Äöaa 4.272.
Step 3: Calculate the Total Area
Finally, calculate the total area of the triangle and the semicircle combined. Use the area formulas for both shapes to derive the total area:
- Area of the triangle: A_triangle = (base ‚àöo height) / 2, hence A_triangle = 12.
- Area of the semicircle: A_semicircle = (œA √o r¬≤) / 2, so substitute the radius.
- Complete the total area calculation: A = A_triangle + A_semicircle = 12 + (9.125œA), expressed in square units.