Question

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 1

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,

$x+5x=360^{\circ}\\ \\6x=360^{\circ}\\ \\x=60^{\circ}.$

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

$\angle BAO=\dfrac{1}{2}(180^{\circ}-60^{\circ})=60^{\circ}.$

Since line CD is tangent to the circle,

$\angle CAO=90^{\circ}.$

Hence,

$\angle CAB=\angle CAO-\angle BAO=90^{\circ}-60^{\circ}=30^{\circ}.$