54. Find x. The ratio of arc AB to arc CA is 10:12. The arc indicated is 140 degrees.

Solution:

• The angle marked 140 degrees is a central angle subtended by arc BC. So, arc BC = $$140^\circ$$.
• The sum of the arcs AB, BC, and CA is 360 degrees.
• Let arc AB = $$10y$$ and arc CA = $$12y$$.
• $$10y + 140^\circ + 12y = 360^\circ$$
• $$22y = 360^\circ - 140^\circ$$
• $$22y = 220^\circ$$
• $$y = 10^\circ$$
• Arc AB = $$10 imes 10^\circ = 100^\circ$$
• Arc CA = $$12 imes 10^\circ = 120^\circ$$
• The angle $$x$$ is an inscribed angle subtended by arc AC.
• $$x = \frac{1}{2} \text{arc AC}$$
• $$x = \frac{1}{2} imes 120^\circ = 60^\circ$$

Answer: $$60^\circ$$