Using the sine and cosine tables, and if needed a calculator, do these problems.

The picture below shows part of a circle:

What is the radius of the circle?

Let us draw a complete circle.

Let the centre be O. Join OA and OB.

We know that,

The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

∴ ∠AOB (External) = 280°

⇒ ∠AOB (Internal) = 360° - 280° = 80°

Let radius be r.

⇒ OA = OB = r.

Draw perpendicular bisector OH on AB

In Δ OHB

OH is a perpendicular bisector of AB.

∴ ∠AOH = ∠BOH , OA = OB and ∠OHA = ∠OHB = 90°

⇒ ∠BOA = ∠BOH + ∠AOH = 80°

⇒ ∠BOA = 2(∠BOH) = 2(∠AOH) = 80°

⇒ ∠BOH = ∠AOH = 40°

In Δ BOA,

Δ BOA is an isosceles triangle (∵ OB = OA)

∴ ∠OBA = ∠OAB = α

Now, ∠OBA + ∠OAB + ∠BOA = 180°

⇒ α + α + 80° = 180°

⇒ 2(α) + 80° = 180°

⇒ α = ∠OBA = 50°

We get,

∠OBA = 50° , ∠BOH = 40° and ∠OHB = 90°

In right triangle BHO,

BH = BO × sin 40°

(From table, sin 40° = 0.642)

⇒ BO = r = 6.23 cm

Radius of a circle is 6.23 cm