The total sum of the arcs in a circle is 360°. For us to be to determine the measures of arc AD, BC and DBC, let's first determine the value of x using the sum of all arcs.
[tex]\text{ Arc AD + Arc DC + Arc CB + Arc BA = 360}^{\circ}[/tex]
We get,
[tex]\text{ Arc AD + Arc DC + Arc CB + Arc BA = 360}^{\circ}[/tex][tex]\text{ (12x) + (17x - 14) + (2x + 5) + 90 = 360}^{\circ}[/tex][tex]\text{ 12x + 17x - 14 + 2x + 5 + 90 = 360}[/tex][tex]\text{ 31x + 81 = 360}[/tex][tex]\text{ 31x = 360 - 81}[/tex][tex]\text{ 31x = 279}[/tex][tex]\text{ }\frac{\text{31x}}{31}\text{ = }\frac{\text{279}}{31}[/tex][tex]\text{ x = 9}^{\circ}[/tex]
Let's use x = 9° to be able to get the measure of the three arcs.
For Arc AD,
Arc AD = 12x = 12(9) = 108°
For Arc BC,
Arc BC = 2x + 5 = 2(9) + 5 = 18 + 5 = 23°
For Arc DBC,
Arc DBC = 12x + 90 + 2x + 5 = 12(9) + 90 + 2(9) + 5 = 108 + 90 + 18 + 5 = 221°
IN SUMMARY:
x = 9°
Arc AD = 108°
Arc BC = 23°
Arc DBC = 221°
