Length of arc = $\frac{\mathrm{\theta }}{360}$ x 2πr

Where r = radius of the circle.

Length of the arc = 50 cm
θ = 75

π = $\frac{22}{7}$

$\frac{75}{360}$ x 2 x $\frac{22}{7}$ x r = 50

$\frac{\mathrm{75\; x\; 2\; x\; 22}}{\mathrm{360\; x\; 7}}$ x r = 50

$\frac{3300}{2520}$ x r = 50

Get rid of the fraction by multiplying both sides of the equation by the denominator (2520)

3300 x r = 50 x 2520
3300 x r = 126000

Divide both sides of the equation by 3300

r = $\frac{126000}{3300}$

r = 38.1818 ≈ 38.2 cm (three significant figures)

1. Find the number of times the cyclist complete one cycle by dividing the distance overed by the circumference.

Number of completed cycles = $\frac{\mathrm{302\; km}}{\mathrm{9\; km}}$ = 33.555

Number of completed cycles = 33

Note: completed cycle is when the cyclist reaches the starting point.

2. Distance from starting point = Distance covered - Number of completed cycles x the circumference

Distance from starting point = 302 km - 33 x 9 km

Distance from starting point = 302 km - 297 km

Distance from starting point = 5 km

Length of the arc = 1$\frac{19}{36}$ cm = $\frac{\mathrm{1\; x\; 36\; +\; 19}}{36}$ cm = $\frac{55}{36}$ cm

π = $\frac{22}{7}$

Radius = 3.5 cm

$\frac{55}{36}$ = $\frac{\mathrm{\theta }}{360}$ x 2 x $\frac{22}{7}$ x 3.5

$\frac{55}{36}$ = $\frac{\mathrm{\theta \; x\; 2\; x\; 22\; x\; 3.5}}{\mathrm{360\; x\; 7}}$

Cross multiply.

36 x θ x 2 x 22 x 3.5 = 55 x 360 x 7

Divide both sides by 36 x 2 x 22 x 3.5

θ = $\frac{\mathrm{55\; x\; 360\; x\; 7}}{\mathrm{36\; x\; 2\; x\; 22\; x\; 3.5}}$

θ = 25°

Length of the arc = $\frac{\mathrm{72°}}{\mathrm{360°}}$ x 2 x $\frac{22}{7}$ x 3.5 cm

Length of the arc = 4.4 cm

Length of the arc = 22 cm

22 = $\frac{\mathrm{\theta }}{360}$ x 2 x $\frac{22}{7}$ x 15

Get rid of the fraction by multiplying both sides by the L.C.M of the denominators 360 and 7. The L.C.M is 360 x 7

22 x 360 x 7 = θ x 2 x 22 x 15

55440 = 660θ

Divide both sides by 660

θ = $\frac{55440}{660}$ = 84°

The angle subtended by the arc is ∠ ZOY

∠ ZOY = 2 x ∠ ZXY

∠ ZOY = 2 x 70° = 140°

Radius (r) = 18 cm

π = $\frac{22}{7}$

Length of arc ZY = $\frac{140}{360}$ x 2 x $\frac{22}{7}$ x 18 cm = 44 cm

Let c = length of the chord.

Method I

Using the cosine rule

c² = 5² + 5² - 2 x 5 x 5 cos70°

c² = 25 + 25 - 50cos70°

c² = 50 - 50cos70°

c² = 32.9

Take the square root of both sides to get rid of the square.

c = $\sqrt{\mathrm{32.9}}$ = 5.7359 cm ≈ 5.7 cm (one decimal place)

Method II

Using the sine rule

Since the radii are the same, the triangle the chord makes with the circle is an isosceles triangle (has two sides equal).

The base angles of an isosceles triangle are the same.

Let y = base angle of the isosceles triangle.

y + y + 70° = 180° (Sum of angles in a triangle).

2y = 180° - 70°

2y = 110°

Divide both sides by 2.

y = $\frac{\mathrm{110°}}{2}$ = 55°

Each of the base angles is 55°

Applying the sine rule,

$\frac{\mathrm{sin\; 70°}}{\mathrm{c}}$ = $\frac{\mathrm{sin\; 55°}}{5}$

Cross multiply.

c x sin 55° = 5 x sin 70°

Divide both sides by sin 55°

c = $\frac{\mathrm{5\; x\; sin\; 70°}}{\mathrm{sin\; 55°}}$ = 5.7358 cm ≈ 5.7 cm (one decimal place)

∠ POR = 2 x ∠ PQR = 2 x 45° = 90°

Radius = 12 cm

π = $\frac{22}{7}$

Length of arc PR = $\frac{\mathrm{90°}}{\mathrm{360°}}$ x 2 x $\frac{22}{7}$ x 12 cm

Length of arc PR = 18.857 cm ≈ 19 cm (nearest cm)

The angle (θ) substended are the same.

Area of a sector = $\frac{\mathrm{\theta }}{360}$ x π x radius x radius

Area = 231 cm²

$\frac{\mathrm{\theta }}{360}$ x π x radius x radius = 231

Make the angle (θ) the subject.

Get rid of the fraction by multiplying both sides by the denominator 360.

360 x $\frac{\mathrm{\theta }}{360}$ x π x radius x radius = 360 x 231

θ x π x radius x radius = 360 x 231

Divide both sides by π x radius x radius

θ = $\frac{\mathrm{360\; x\; 231}}{\mathrm{\pi \; x\; radius\; x\; radius}}$ --- eqn 1

Length of an arc = $\frac{\mathrm{\theta }}{360}$ x 2 x π x radius

Length = 66 cm

$\frac{\mathrm{\theta }}{360}$ x 2 x π x radius = 66

Make the angle (θ) the subject.

Get rid of the fraction by multiplying both sides by the denominator, 360.

360 x $\frac{\mathrm{\theta }}{360}$ x 2 x π x radius = 360 x 66

θ x 2 x π x radius = 360 x 66

Divide both sides by 2 x π x radius

θ = $\frac{\mathrm{360\; x\; 66}}{\mathrm{2\; x\; \pi \; x\; radius}}$ --- eqn 2

--- eqn 1 = --- eqn 2

$\frac{\mathrm{360\; x\; 231}}{\mathrm{\pi \; x\; radius\; x\; radius}}$ = $\frac{\mathrm{360\; x\; 66}}{\mathrm{2\; x\; \pi \; x\; radius}}$

Cross multiply.

360 x 231 x 2 x π x radius = 360 x 66 x π x radius x radius

Divide both sides by 360 x π x radius to cancel out.

$\frac{\mathrm{360\; x\; 231\; x\; 2\; x\; \pi \; x\; radius}}{\mathrm{360\; x\; \pi \; x\; radius}}$ = $\frac{\mathrm{360\; x\; 66\; x\; \pi \; x\; radius\; x\; radius}}{\mathrm{360\; x\; \pi \; x\; radius}}$

231 x 2 = 66 x radius

Divide both sides by 66

Radius = $\frac{\mathrm{231\; x\; 2}}{66}$ = 7 cm

Length of an arc = $\frac{\mathrm{\theta }}{360}$ x 2 x π x radius

Length of the arc = 11 cm

θ = 60

π = $\frac{22}{7}$

Substitute the values into the equation and solve for the radius.

$\frac{60}{360}$ x 2 x $\frac{22}{7}$ x radius = 11

Get rid of the fraction by multiplying both sides by the L.C.M of the denominators, 360 and 7 or (360 x 7).

360 x 7 x $\frac{60}{360}$ x 2 x $\frac{22}{7}$ x radius = 11 x 360 x 7

60 x 2 x 22 x radius = 11 x 360 x 7

Divide both sides by 60 x 2 x 22

radius = $\frac{\mathrm{11\; x\; 360\; x\; 7}}{\mathrm{60\; x\; 2\; x\; 22}}$

radius = 10.5 cm = 10$\frac{1}{2}$ cm