Given: k(o, r), ab = bc
m∠b =120°, r=14cm
find: ab

AB=14 cm

Step-by-step explanation:

step 1

Find the measure of major arc AC

we know that

The inscribed angle is half that of the arc it comprises.

so

m∠B=(1/2)[major arc AC]

we have

m∠B=120°

substitute

120°=(1/2)[major arc AC]

240°=major arc AC

so

major arc AC=240°

step 2

Find the measure of arc ABC

we know that

arc AC+arc ABC=360°

substitute

240°+arc ABC=360°

arc ABC=120°

step 3

Find the measure of angle AOC  

m∠AOC=arc ABC=120° > by central angle

so

The triangle AOC is an isosceles triangle

OA=OC=14 cm > is the radius

The internal angles of triangle AOC are

m∠CAO=m∠OCA=30°

The triangle ABC is an isosceles triangle

AB=BC

The internal angles of triangle ABC are

m∠BAC=m∠ACB=30°

so

Triangles AOC and ABC are congruent by ASA similarity postulate ( two angles and included side)

therefore

AO=AB=14 cm

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