Question 141
A figura representa um mapa com a localização de três cidades: A, B e C. As distâncias entre as cidades, em linha reta, são: A a B = 80 km, B a C = 60 km e A a C = 100 km.
[Figure]
O ângulo formado pelas estradas AB e AC, em graus, é
(A) $30^\circ$ (B) $37^\circ$ (C) $45^\circ$ (D) $53^\circ$ (E) $60^\circ$

18. If the three interior angles of $\triangle A B C$ satisfy $\sin A : \sin B : \sin C = 5 : 11 : 13$ , then $\triangle A B C$
A. must be an acute triangle
B. must be a right triangle
C. must be an obtuse triangle
D. could be either an acute triangle or an obtuse triangle

III. Solution Problems (Total Score: 74 points)

In $\triangle A B C$ , $\cos C = \frac { 2 } { 3 } , A C = 4 , B C = 3$ , then $\cos B =$
A. $\frac { 1 } { 9 }$
B. $\frac { 1 } { 3 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 2 } { 3 }$

In $\triangle ABC$, $BC = 2$, $AC = 1 + \sqrt{3}$, $AB = \sqrt{6}$, then $A = $ ( )
A. $45°$
B. $60°$
C. $120°$
D. $135°$

25. In a rhombus, the geometric mean of the two diagonals is the length of the rhombus's side. The smaller angle of the triangle formed by drawing the diagonals of this rhombus is how many degrees?
(1) $10$ (2) $15$ (3) $30$ (4) $45$

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

Let ABC be a triangle such that $\angle \mathrm { ACB } = \frac { \pi } { 6 }$ and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively. The value(s) of x for which $\mathrm { a } = \mathrm { x } ^ { 2 } + \mathrm { x } + 1 , \mathrm {~b} = \mathrm { x } ^ { 2 } - 1$ and $\mathrm { c } = 2 \mathrm { x } + 1$ is (are)
A) $- ( 2 + \sqrt { 3 } )$
B) $1 + \sqrt { 3 }$
C) $2 + \sqrt { 3 }$
D) $4 \sqrt { 3 }$

If in a triangle $ABC$, $\frac{b+c}{11} = \frac{c+a}{12} = \frac{a+b}{13}$, then $\cos A$ is equal to
(1) $5/7$
(2) $1/5$
(3) $35/19$
(4) $19/35$

In a triangle $\mathrm { ABC } , \mathrm { BC } = 7 , \mathrm { AC } = 8 , \mathrm { AB } = \alpha \in \mathrm { N }$ and $\cos \mathrm { A } = \frac { 2 } { 3 }$. If $49 \cos ( 3 \mathrm { C } ) + 42 = \frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$

From 6, 8, 10, 12, select any three distinct numbers as the three sides of a triangle, and let $\theta$ be the largest interior angle of this triangle. Among all possible triangles formed, the minimum value of $\cos \theta$ is (Express as a fraction in lowest terms)

The triangle ABC is drawn on unit squares as shown above. What is the tangent of angle $B$?
A) $\frac { 25 } { 4 }$
B) $\frac { 34 } { 5 }$
C) $\frac { 40 } { 9 }$
D) 4
E) 5

ABCD is a square, $|BE| = 5$ cm, $|EC| = 7$ cm, $m(\widehat{EAC}) = x$.
According to the given information, what is $\tan x$?
A) $\frac { 4 } { 13 }$
B) $\frac { 6 } { 13 }$
C) $\frac { 9 } { 13 }$
D) $\frac { 5 } { 17 }$
E) $\frac { 7 } { 17 }$

$| \mathrm { AB } | = 6$ units $| \mathrm { BH } | = 2$ units $[ \mathrm { BC } ] \cap [ \mathrm { GF } ] = \mathrm { H }$ $\mathrm { m } ( \widehat { \mathrm { GHC } } ) = \mathrm { x }$
In the figure, $ABCD$ and $AEFG$ are congruent squares. Accordingly, what is the value of $\tan ( x )$?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 5 } { 4 }$

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

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