8. At two points $A$ and $B$ that are 2 kilometers apart, target $C$ is measured. If $\angle CAB = 75°, \angle CBA = 60°$, then the distance between points $A$ and $C$ is $\_\_\_\_$ kilometers.

In $\triangle ABC$, $a = 3$, $b = \sqrt { 6 }$, $\angle A = \frac { 2 \pi } { 3 }$, then $\angle B =$

137- The internal bisector of angle $A$ in triangle $ABC$ divides the opposite side into segments of $3/5$ and $2/5$ units. If the measure of angle $C$ is $60$ degrees, the smaller side of the triangle is how many units?
\[ \text{(1)}\ 3/75 \qquad \text{(2)}\ 4/25 \qquad \text{(3)}\ 4/75 \qquad \text{(4)}\ 5/25 \]

10. The triangle $P Q R$ is inscribed in the circle $x 2 + y 2 = 25$. If $Q$ and $R$ have coordinates ( 3 , $4 )$ and $( - 4,3 )$ respectively, then $\angle P Q R$ is equal to:
(A) $\pi / 2$
(B) $\pi / 3$
(C) $\pi / 4$
(D) $\pi / 6$

8. If the angles of a triangle are in the ratio $4 : 1 : 1$, then the ratio of the longest side to the perimeter is:
(a) $\sqrt { } 3 : ( 2 + \sqrt { } 3 )$
(b) $1 : 6$
(c) $1 : 2 + \sqrt { } 3$
(d) $2 : 3$

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Maths \& Computer Science and Computer Science applicants should turn to page 14. [Figure] [Figure]
A triangle $A B C$ has sides $B C , C A$ and $A B$ of sides $a , b$ and $c$ respectively, and angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$ where $0 \leqslant \alpha , \beta , \gamma \leqslant \frac { 1 } { 2 } \pi$.
(i) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$.
Deduce the sine rule
$$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$
(ii) The points $P , Q$ and $R$ are respectively the feet of the perpeniculars from $A$ to $B C$, $B$ to $C A$, and $C$ to $A B$ as shown.
Prove that
$$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$
(iii) For what triangles $A B C$, with angles $\alpha , \beta , \gamma$ as above, does the equation
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$
hold?

5. Assume that the distances between towns A, B, and C are all equal to 20 kilometers. Two straight roads intersect at town D, one passing through towns A and B, and the other passing through town C. On an accurately scaled map, the angle between the two roads is measured to be $45^{\circ}$. Then the distance between towns C and D is approximately
(1) 24.5 kilometers
(2) 25 kilometers
(3) 25.5 kilometers
(4) 26 kilometers
(5) 26.5 kilometers

ABC is a triangle
$$\begin{aligned} & \mathrm { m } ( \widehat { \mathrm { ABC } } ) = 50 ^ { \circ } \\ & \mathrm { m } ( \widehat { \mathrm { CAB } } ) = 100 ^ { \circ } \end{aligned}$$
According to the given information, the expression $\frac { | a - b | + | b - c | + | c - a | } { 2 }$ is equal to which of the following?
A) a-c
B) $a - b$
C) $b - c$
D) $b - a$
E) $\mathrm { c } - \mathrm { b }$

ABC and DEC are triangles
$$\begin{aligned} & \mathrm { m } ( \widehat { \mathrm { CAB } } ) = \mathrm { m } ( \widehat { \mathrm { DEC } } ) \\ & | \mathrm { AD } | = 5 \mathrm {~cm} \\ & | \mathrm { DC } | = 3 \mathrm {~cm} \\ & | \mathrm {~EB} | = 2 \mathrm {~cm} \\ & | \mathrm { BC } | = x \end{aligned}$$
According to the given information, what is x in cm?
A) 4
B) 5
C) $\frac { 9 } { 2 }$
D) $\frac { 10 } { 3 }$
E) $\frac { 13 } { 3 }$