Chord AB subtends two arcs with measures in the ratio of 1:5. Line l is tangent to a circle at point A. Find the measure of the angle between the tangent and secant AB .

Answer:30°Step-by-step explanation:If chord AB subtends two arcs with measures in the ratio of 1:5, then the measure of minor arc is x and the measure of major arc is 5x. Thus,[tex]x+5x=360^{\circ}\\ \\6x=360^{\circ}\\ \\x=60^{\circ}.[/tex]Thus, the measure of the angle AOB  is 60°. Consider isosceles triangle AOB (because AO=BO=radius of the circle). The angles adjacent to the base AB are congruent, thus[tex]\angle BAO=\dfrac{1}{2}(180^{\circ}-60^{\circ})=60^{\circ}.[/tex]Since line CD is tangent to the circle, [tex]\angle CAO=90^{\circ}.[/tex]Hence,[tex]\angle CAB=\angle CAO-\angle BAO=90^{\circ}-60^{\circ}=30^{\circ}.[/tex]