On a globe with center $O$, there are five points $A$, $B$, $C$, $D$, $E$. Points $A$, $B$, $C$ are all on the equator with longitudes of East $0^{\circ}$, $60^{\circ}$, and $90^{\circ}$ respectively. Points $D$ and $E$ are both on the $30^{\circ}$ North latitude line with longitudes of East $0^{\circ}$ and $180^{\circ}$ respectively. Select the correct options.
(1) The length of the equator equals the sum of the lengths of the meridians at East $0^{\circ}$ and $180^{\circ}$ (2) The length of the $45^{\circ}$ North latitude line equals $\frac{1}{2}$ of the equator's length (3) The shortest path length from $A$ to $B$ along the equator equals the path length from $D$ to the North Pole along the East $0^{\circ}$ meridian (4) The path length from $D$ to $E$ along the $30^{\circ}$ North latitude line equals the sum of the path lengths from $D$ to the North Pole along the East $0^{\circ}$ meridian and from the North Pole to $E$ along the East $180^{\circ}$ meridian (5) The line passing through the North Pole and point $A$ is perpendicular to the line passing through the North Pole and point $C$

ABCD is a rectangle $\wideparen { \mathrm { CE } }$ is a circular arc with center A $| \mathrm { DA } | = 4 \mathrm {~cm}$ $| \mathrm { AC } | = 8 \mathrm {~cm}$
According to the given information, what is the area of the shaded circular sector in $\mathbf { cm } ^ { \mathbf { 2 } }$?
A) $\frac { 16 \pi } { 3 }$
B) $\frac { 20 \pi } { 3 }$
C) $\frac { 25 \pi } { 3 }$
D) $\frac { 28 \pi } { 3 }$
E) $\frac { 32 \pi } { 3 }$

O is the center of the circle
AT is tangent to the circle at point T
$$\begin{aligned} & | A T | = 3 \mathrm {~cm} \\ & \mathrm {~m} ( \widehat { \mathrm { OAT } } ) = 45 ^ { \circ } \end{aligned}$$
According to the given information, what is the length of arc BT in cm?
A) $\frac { \pi } { 2 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 3 \pi } { 4 }$
D) $\frac { 4 \pi } { 5 }$
E) $\frac { 5 \pi } { 6 }$

$$|\widehat{AD}| = a \text{ units}$$ $$|\widehat{BC}| = b \text{ units}$$ $$|DC| = c \text{ units}$$
The OAD and OBC circular sectors with center O are given above.
Accordingly, the area of the shaded region is equal to which of the following in terms of $a, b$ and $c$?
A) $\frac{(a + b) \cdot c}{2}$ B) $\frac{(b - a) \cdot c}{2}$ C) $\frac{2(a + b)}{c}$ D) $\frac{2(b - a)}{c}$ E) $\frac{a \cdot b \cdot c}{2}$

Below is a running park consisting of an isosceles right triangle and a semicircle with the hypotenuse of this triangle as its diameter. There are three running paths in this park. Ay\c{c}a, Bar\i\c{s}, and Cem start running from the starting point at the same time using paths A, B, and C respectively and reach the finish point.
Given that the speeds of Ay\c{c}a, Bar\i\c{s}, and Cem are 4 km, $\mathbf { 2 ~ km }$, and $\mathbf { 3 ~ km }$ per hour respectively, what is the order of arrival at the finish point?
A) Ay\c{c}a, Bar\i\c{s}, Cem
B) Ay\c{c}a, Cem, Bar\i\c{s}
C) Bar\i\c{s}, Cem, Ay\c{c}a
D) Bar\i\c{s}, Ay\c{c}a, Cem
E) Cem, Ay\c{c}a, Bar\i\c{s}

In an amusement park, there is a circular Ferris wheel on flat ground consisting of identical cabins as shown in the figure, rotating in only one direction. A person boards one cabin when the cabin is closest to the ground.
Meryem boards a cabin and after the Ferris wheel rotates $48°$, Nisa also boards a cabin.
Accordingly, after Nisa boards her cabin, when the heights of the cabins where Meryem and Nisa are located are equal for the first time, how many degrees has the Ferris wheel rotated?
A) 130
B) 138
C) 144
D) 150
E) 156

Ayşe and Ferhat enter a store to buy pizza. From a whole pizza divided into 13 circular slices in this store; the 2 slices that Ayşe buys are identical to each other, while the 11 slices that Ferhat buys are also identical to each other.
Later, they combine three of these slices to obtain a semicircular pizza.
Accordingly, what is the measure of the central angle of one of the larger slices in degrees?
A) 90
B) 81
C) 75
D) 72
E) 60

O is the center of a semicircle, ABCD is a rectangle.
A, O and B are collinear.
$|AE| = |ED| = \dfrac{1}{2}$ unit, $m(\widehat{AOE}) = x$
Points C and D lie on the semicircle with center O.
What is the length of a diagonal of rectangle ABCD in terms of $x$?
A) $\tan x$ B) $\operatorname{cosec} x$ C) $\sec x$ D) $\sin x$ E) $\cos x$

For each pizza at a restaurant, a circular pizza dough is first rolled out. Then, an orange-colored ingredient section is created to form a concentric circle with this dough. The top view of the pizza ordered by Ali and Ayşe at this restaurant is shown in the figure.
Ayşe divides the pizza shown in the figure into 8 equal slices; she takes 3 of these slices and gives 5 to Ali. While Ali eats all of his slices, Ayşe eats only the orange-colored portions of her slices as shown in the figure.
In the end, the area of the portions Ali ate is calculated to be 2.4 times the area of the portions Ayşe ate.
What is the ratio of the radius of this pizza to the radius of the orange section?
A) $\dfrac{3}{2}$ B) $\dfrac{4}{3}$ C) $\dfrac{5}{4}$ D) $\dfrac{6}{5}$ E) $\dfrac{7}{6}$

A wire in the shape of a circular arc with a central angle of $120^{\circ}$ has its ends mounted on the top edge of a rectangular board. This board is hung on a nail in the wall as shown in Figure 1, with the midpoint of the wire aligned with the nail, so that the long sides of the board are parallel to the ground.
Then this wire is removed, bent into a semicircle shape, and mounted on the board again. The board is hung on the same nail as shown in Figure 2, with the midpoint of the wire again aligned with the nail, so that the long sides of the board are again parallel to the ground. As a result, the height of the board from the ground decreases by 8 units compared to the initial position.
What is the length of the wire?
A) $24\pi$ B) $28\pi$ C) $32\pi$ D) $36\pi$ E) $40\pi$