51. Find x. The ratio of arc AB to arc BC to arc AC is 7:11:6.

Solution:

• The sum of the arcs AB, BC, and AC is 360 degrees.
• Let the arcs be $$7x$$, $$11x$$, and $$6x$$.
• $$7x + 11x + 6x = 360^\circ$$
• $$24x = 360^\circ$$
• $$x = \frac{360^\circ}{24} = 15^\circ$$
• Arc AB = $$7 \times 15^\circ = 105^\circ$$
• Arc BC = $$11 \times 15^\circ = 165^\circ$$
• Arc AC = $$6 imes 15^\circ = 90^\circ$$
• The angle $$x$$ is the inscribed angle subtended by arc BC.
• $$x = \frac{1}{2} \text{arc BC}$$
• $$x = \frac{1}{2} imes 165^\circ = 82.5^\circ$$

Answer: $$82.5^\circ$$