Question

The difference of measures between the arcs subtended by chord AB is 160°. Line l is tangent to the circle at point A. Find the measure of the angle between the tangent l and secant AB.

Answer 1

50°

Explanation:

x + (x + 160) = 360

2x = 200

x = 100

OAB = OBA = (180 - 100)/2

= 80/2

= 40

Required angle:

90 - 40 = 50

Answer 2

Given:

Given that the difference of measures between the arcs subtended by chord AB is 160°.

Line l is tangent to the circle at point A.

We need to determine the measure of the angle between the tangent l and secant AB.

Measure of arc AB:

Let x denote the measure of minor arc AB.

Thus, we have;

`x+(x+160^(\circ))=360^(\circ)`

`2x+160^(\circ)=360^(\circ)`

`2x=200^(\circ)`

`x=100^(\circ)`

Thus, the measure of the minor arc AB is 100°

Measure of angle between the tangent l and the secant AB:

Since, we know the property that, "if a tangent and the chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc".

Thus, we have;

Measure of angle =
`(1)/(2) m( \widehat{AB})`

Substituting the value, we get;

Measure of angle =
`(1)/(2) (100^(\circ)) = 50^(\circ)`

Thus, the measure of angle between the tangent l and the secant AB is 50°