ABCD is a square, $\mathrm { AE } \cap \mathrm { BF } = \{ \mathrm { G } \}$, $| \mathrm { BC } | = 6$ units, $| \mathrm { DE } | = 4$ units, $| \mathrm { AF } | = 3$ units, $\mathrm { m } ( \widehat { \mathrm { FGE } } ) = \mathrm { x }$.
According to the given information above, what is the value of $\cot ( x )$?
A) $\frac { - 1 } { 4 }$
B) $\frac { - 5 } { 4 }$
C) $\frac { - 3 } { 8 }$
D) $\frac { - 1 } { 8 }$
E) $\frac { - 5 } { 8 }$

ABCD rectangle, DEFG square\ $| \mathrm { DE } | = 6$ units\ $| \mathrm { AE } | = 3$ units\ $| \mathrm { AB } | = 12$ units\ $\mathrm { m } \widehat { ( \mathrm { BFC } ) } = \mathrm { x }$
Accordingly, what is $\cot ( x )$?\ A) $\frac { 1 } { \sqrt { 2 } }$\ B) $\frac { 1 } { 3 }$\ C) 1\ D) $\sqrt { 3 }$\ E) 2

$ABC$ triangle, AFD equilateral triangle, $[DE]$ // $[AB]$, $m ( \widehat { DFC } ) = x$
In the figure, $m \widehat { ( \mathrm { ACF } ) } = m \widehat { ( \mathrm { FCB } ) } = m \widehat { ( \mathrm { DEC } ) }$ and points $D$, $E$, $F$ lie on the sides of triangle ABC.
Accordingly, what is x in degrees?\ A) 20\ B) 25\ C) 30\ D) 35\ E) 40

ABC is a triangle $$\begin{aligned}& | \mathrm { AD } | = | \mathrm { CD } | = | \mathrm { BC } | \\& \mathrm { m } ( \widehat { \mathrm { BAD } } ) = 20 ^ { \circ } \\& \mathrm { m } ( \widehat { \mathrm { BCD } } ) = 60 ^ { \circ } \\& \mathrm { m } ( \widehat { \mathrm { ACD } } ) = \mathrm { x }\end{aligned}$$ Accordingly, what is x in degrees?\ A) 5\ B) 10\ C) 15\ D) 20\ E) 25

ABC isosceles triangle\ $AD \cap BC = \{ E \}$\ $AD \perp BC$\ $| \mathrm { AB } | = | \mathrm { BD } | = 6$ units\ $| AC | = | BC | = 9$ units\ $| CE | = \mathrm { x }$
Accordingly, what is x in units?\ A) 4\ B) 5\ C) 6\ D) 7\ E) 8

$ABC$ is a right triangle\ $\mathrm { AB } \perp \mathrm { AC }$\ $\mathrm { DE } \perp \mathrm { BC }$\ $| \mathrm { AD } | = | \mathrm { DB } | = 3$ units\ In triangle $ABC$, $D$ and $E$ lie on sides $AB$ and $BC$ respectively.\ If the area of triangle $ABC$ is 6 times the area of triangle $BDE$, what is $| AC |$ in units?\ A) $2 \sqrt { 3 }$\ B) $3 \sqrt { 2 }$\ C) $2 \sqrt { 6 }$\ D) 3\ E) 6

ABC and BDE are equilateral triangles\ $[ \mathrm { BD } ] \perp [ \mathrm { AC } ]$\ $[ \mathrm { BF } ] \perp [ \mathrm { DE } ]$\ $[ \mathrm { FH } ] \perp [ \mathrm { BE } ]$\ $| \mathrm { AB } | = 16$ units
Accordingly, what is the area of triangle BFH in square units?\ A) $12 \sqrt { 3 }$\ B) $15 \sqrt { 3 }$\ C) $18 \sqrt { 3 }$\ D) $20 \sqrt { 3 }$\ E) $24 \sqrt { 3 }$

$ABC$ right triangle\ $[ \mathrm { AC } ] \perp [ \mathrm { BC } ]$\ $[AB]$ // $[DE]$\ $[BC]$ // $[FH]$\ $| \mathrm { AD } | = | \mathrm { DH } | = | \mathrm { HC } |$\ $| \mathrm { GE } | = 4$ units\ $| \mathrm { GF } | = 2$ units
Accordingly, what is the area of triangle ABC in square units?\ A) $9 \sqrt { 3 }$\ B) $12 \sqrt { 3 }$\ C) $15 \sqrt { 3 }$\ D) $18 \sqrt { 3 }$\ E) $20 \sqrt { 3 }$

ABCD right trapezoid, ABD equilateral triangle\ $[AB]$ // $[DC]$\ $| \mathrm { BF } | = 4 | \mathrm { DF } |$\ $| \mathrm { AB } | = 8$ units
Accordingly, what is the area of right trapezoid ABCE in square units?\ A) $10 \sqrt { 3 }$\ B) $12 \sqrt { 3 }$\ C) $16 \sqrt { 3 }$\ D) $18 \sqrt { 3 }$\ E) $20 \sqrt { 3 }$

ABCD kite\ $[ \mathrm { AC } ] \perp [ \mathrm { BD } ]$\ $| \mathrm { AB } | = | \mathrm { BC } |$\ $| \mathrm { AD } | = | \mathrm { DC } |$\ $| \mathrm { BE } | = 4 | \mathrm { ED } |$\ $| \mathrm { AC } | = 16$ units
The area of kite ABCD in the figure is 160 square units.\ Accordingly, what is the perimeter of kite ABCD in units?\ A) $20 \sqrt { 5 }$\ B) $24 \sqrt { 5 }$\ C) $28 \sqrt { 5 }$

Captain Temel will take the tourists on his boat from island A to island B in the morning, from island B to island C at noon, and from island C to island A in the evening.
The points where the boat will dock at the islands are marked as the vertices of a triangle ABC where side AB equals side BC, as shown in the figure.
Since Captain Temel knows he will travel in the dark on the return journey, as he travels from A to B and from B to C, he notes on a piece of paper the angle between the compass needle pointing north and the path he follows.
Accordingly, how should Captain Temel set his compass to go from C to A?

A triangular ABC cardboard with vertices labeled with letters A, B, and C is shown as in Figure 1. 3 ABC cardboards can be assembled on a flat surface as shown in Figure 2 by overlapping the A vertices and leaving no gaps between the edges and without the cardboards overlapping.
The same process can be done using 9 ABC cardboards by overlapping the B vertices.
Accordingly, using how many ABC cardboards can this process be done by overlapping the C vertices?
A) 10
B) 12
C) 15
D) 18
E) 20

A seesaw on a flat ground as shown in Figure 1 consists of a straight segment 30 units long and a straight support 9 units long located at the exact center of this segment.
As shown in Figure 2, when the left end of the seesaw touches the ground, a shaded region in the shape of a right trapezoid is formed on the right side.
Accordingly, what is the perimeter of this trapezoid in units?
A) 54
B) 55
C) 56
D) 57
E) 58

In the figure, a regular hexagon and a square sharing one side are given. A regular polygon sharing one side with the regular hexagon and one side with the square is to be drawn as shown in the figure.
Accordingly, how many sides does the regular polygon to be drawn have?
A) 10
B) 12
C) 15
D) 16
E) 18

3 identical isosceles trapezoids are joined together such that any two of them share a vertex as shown below.
One side of the large triangle formed is 6 units, and one side of the small triangle is 3 units.
Accordingly, what is the perimeter of one of these isosceles trapezoids in units?
A) 10
B) 10.5
C) 11
D) 11.5
E) 12

In the square-shaped paper ABCD given in Figure 1, $|\mathrm{DE}| = 6$ and $|\mathrm{BF}| = 9$ units. When this paper is folded along the line segments $[\mathrm{CE}]$ and $[\mathrm{CF}]$ as shown in the figure, the BC side and DC side of the square coincide as shown in Figure 2.
Accordingly, what is the perimeter of square ABCD in units?
A) 64
B) 68
C) 72
D) 76
E) 80

One interior angle of an $n$-sided regular polygon is calculated as $\frac{(n-2) \cdot 180^{\circ}}{n}$.
A triangular piece of paper is cut along the dashed lines as shown in the figure, 3 triangular pieces are removed, and a regular hexagon is obtained.
Given that the sum of the perimeters of the removed triangles is 36 units, what is the perimeter of the hexagon?
A) 18
B) 24
C) 30
D) 36
E) 42

Ali places the sharp end of a compass on a point on paper and, without changing the compass opening, draws a circle with a diameter of 21 cm.
Given that the lengths of the compass legs are 7.5 and 12 cm, what is the measure of the angle between the compass legs in degrees?
A) 30
B) 45
C) 60
D) 90
E) 120

In a triangle, one interior angle measure equals the average of the measures of the other two interior angles. The shortest and longest sides of this triangle are 10 and 16 units long, respectively.
Accordingly, what is the length of the third side of this triangle in units?
A) 11 B) 12 C) 13 D) 14 E) 15

In a triangle $ABC$, the length of side $AB$ is equal to half the length of side $BC$.
If two of the altitudes of this triangle have lengths 4 units and 10 units, which of the following could be the length of the other altitude?
I. 2 units II. 5 units III. 8 units
A) Only I B) Only II C) Only III D) I and II E) II and III