Question

Can someone explain this to me please

Answer 1

Answer:

c. 36·x

Step-by-step explanation:

Part A

The details of the circle are;

The area of the circle, A = 12·π cm²

The diameter of the circle, d = $\overline {AB}$

Given that $\overline {AB}$ is the diameter of the circle, we have;

The length of the arc AB = Half the the length of the circumference of the circle

Therefore, we have;

A = 12·π = π·d²/4 = π·$\overline {AB}$²/4

Therefore;

12 = $\overline {AB}$²/4

4 × 12 = $\overline {AB}$²

$\overline {AB}$² = 48

$\overline {AB}$ = √48 = 4·√3

$\overline {AB}$ = 4·√3

The circumference of the circle, C = π·d = π·$\overline {AB}$

Arc AB = Half the the length of the circumference of the circle = C/2

Arc AB = C/2 = π·$\overline {AB}$/2

$\overline {AB}$ = 4·√3

∴ C/2 = π·4·√3/2 = 2·√3·π

The length of arc AB = 2·√3·π cm

Part B

The given parameters are;

The length of $\overline {OF}$ = The length of $\overline {FB}$

Angle D = angle B

The radius of the circle = 6·x

The measure of arc EF = 60°

The required information = The perimeter of triangle DOB

We have;

Given that the base angles of the triangles DOB are equal, we have that ΔDOB is an isosceles triangle, therefore;

The length of $\overline {OD}$ = The length of $\overline {OB}$

The length of $\overline {OB}$ = $\overline {OF}$ + $\overline {FB}$ = $\overline {OF}$ + $\overline {OF}$ = 2 × $\overline {OF}$

∴ The length of $\overline {OD}$ = 2 × $\overline {OF}$ = The length of $\overline {OB}$

Given that arc EF = 60°, and the point 'O' is the center of the circle, we have;

∠EOF = The measure of arc EF = 60° = ∠DOB

Therefore, in ΔDOB, we have;

∠D + ∠B = 180° - ∠DOB = 180° - 60° = 120°

∵ ∠D = ∠B, we have;

∠D + ∠B = ∠D + ∠D = 2 × ∠D = 120°

∠D = ∠B = 120°/2 = 60°

All three interior angles of ΔDOB = 60°

∴ ΔDOB is an equilateral triangle and all sides of ΔDOB are equal

Therefore;

The length of $\overline {OD}$ = The length of $\overline {OB}$ = The length of $\overline {DB}$  = 2 × $\overline {OF}$

The perimeter of ΔDOB = The length of $\overline {OD}$ + The length of $\overline {OB}$ + The length of $\overline {DB}$ = 2 × $\overline {OF}$ + 2 × $\overline {OF}$ + 2 × $\overline {OF}$ = 6 × $\overline {OF}$

∴ The perimeter of ΔDOB = 6 × $\overline {OF}$

The radius of the circle = $\overline {OF}$ = 6·x

∴ The perimeter of ΔDOB = 6 × 6·x = 36·x