To solve the problem of finding the four angles of an inscribed quadrilateral where the vertices divide the circle in the ratio [tex]\(1: 2: 5: 4\)[/tex], let’s proceed step by step:
1. Understanding the Ratio and Total Parts: First, let's recognize that the circle is divided into parts by the vertices in the ratio [tex]\(1:2:5:4\)[/tex]. Adding these parts together, we get:
[tex]\[
1 + 2 + 5 + 4 = 12
\][/tex]
So, the circle's circumference is divided into 12 equal parts.
2. Finding Individual Angles: The sum of a circle's angles is [tex]\(360^\circ\)[/tex]. We can determine each angle corresponding to each part by multiplying the ratio by [tex]\(\frac{360^\circ}{12}\)[/tex].
- First Part ([tex]\(1\)[/tex] part out of [tex]\(12\)[/tex]):
[tex]\[
\left(\frac{1}{12} \right) \times 360^\circ = 30.0^\circ
\][/tex]
- Second Part ([tex]\(2\)[/tex] parts out of [tex]\(12\)[/tex]):
[tex]\[
\left(\frac{2}{12} \right) \times 360^\circ = 60.0^\circ
\][/tex]
- Third Part ([tex]\(5\)[/tex] parts out of [tex]\(12\)[/tex]):
[tex]\[
\left(\frac{5}{12} \right) \times 360^\circ = 150.0^\circ
\][/tex]
- Fourth Part ([tex]\(4\)[/tex] parts out of [tex]\(12\)[/tex]):
[tex]\[
\left(\frac{4}{12} \right) \times 360^\circ = 120.0^\circ
\][/tex]
3. Assigning the Angles: Now we have determined the angles, which are [tex]\(30.0^\circ\)[/tex], [tex]\(60.0^\circ\)[/tex], [tex]\(150.0^\circ\)[/tex], and [tex]\(120.0^\circ\)[/tex].
Thus, the four angles of the quadrilateral are [tex]\(30.0^\circ\)[/tex], [tex]\(60.0^\circ\)[/tex], [tex]\(150.0^\circ\)[/tex], and [tex]\(120.0^\circ\)[/tex].
To fill in the drop-down menus correctly, select:
- The first menu: [tex]\(30.0\)[/tex]
- The second menu: [tex]\(,\)[/tex]
- The third menu: [tex]\(60.0\)[/tex]
- The fourth menu: [tex]\(,\)[/tex]
- The fifth menu: [tex]\(150.0\)[/tex]
- The sixth menu: [tex]\(\text{}^\circ\)[/tex]
- The seventh menu: [tex]\(120.0\)[/tex]
So, the answer is: [tex]\(30.0^\circ\)[/tex], [tex]\(60.0^\circ\)[/tex], [tex]\(150.0^\circ\)[/tex], and [tex]\(120.0^\circ\)[/tex].
