To determine the ratio of the areas of two sectors in a semicircle whose angles are in the ratio 4:5, let's go through the problem step-by-step:
1. Understand the total angle in a semicircle:
A semicircle spans an angle of [tex]\(180^\circ\)[/tex].
2. Determine the angles of the two sectors:
The angles of the two sectors are in the ratio 4:5. This means if we divide [tex]\(180^\circ\)[/tex] into parts according to this ratio, we need to find how much each part represents.
3. Calculate the angles of each sector:
Let's denote the total number of parts as the sum of the ratio numbers, i.e., [tex]\(4 + 5 = 9\)[/tex] parts. So, each part corresponds to [tex]\(\frac{180}{9} = 20^\circ\)[/tex].
Therefore:
- The angle of the first sector (with a ratio of 4 parts) is [tex]\(4 \times 20^\circ = 80^\circ\)[/tex].
- The angle of the second sector (with a ratio of 5 parts) is [tex]\(5 \times 20^\circ = 100^\circ\)[/tex].
4. Relate the areas of sectors to their angles:
The area of a sector of a circle (or semicircle in this case) is directly proportional to its angle. That means the ratio of the areas of the two sectors will be the same as the ratio of their angles.
5. Finding the ratio of the areas:
Since the ratio of their angles is [tex]\(80^\circ : 100^\circ = 4: 5\)[/tex], the ratio of their areas will also be [tex]\(4: 5\)[/tex].
Thus, the ratio of the areas of the two sectors is
[tex]\[ \boxed{4: 5} \][/tex]
So, the correct option is:
2) [tex]\(4: 5\)[/tex]
