In each of the following independent situations we want to construct a triangle $ABC$ satisfying the given conditions. In each case state how many such triangles $ABC$ exist up to congruence.
(i) $AB = 30 \quad BC = 95 \quad AC = 55$
(ii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad \angle C = 55 ^ { \circ }$
(iii) $\angle A = 30 ^ { \circ } \quad \angle B = 95 ^ { \circ } \quad BC = 55$
(iv) $\angle A = 30 ^ { \circ } \quad AB = 95 \quad BC = 55$

We want to construct a triangle ABC such that angle A is $20.21 ^ { \circ }$, side AB has length 1 and side BC has length $x$ where $x$ is a positive real number. Let $N ( x ) =$ the number of pairwise noncongruent triangles with the required properties.
(a) There exists a value of $x$ such that $N ( x ) = 0$.
(b) There exists a value of $x$ such that $N ( x ) = 1$.
(c) There exists a value of $x$ such that $N ( x ) = 2$.
(d) There exists a value of $x$ such that $N ( x ) = 3$.

The number of triplets $( a , b , c )$ of integers such that $a < b < c$ and $a , b , c$ are sides of a triangle with perimeter 21 is
(a) 7 .
(B) 8.
(C) 11 .
(D) 12 .

10. Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $A B C$ ( $R$ being the radius of the circumcircle)?
(A) $a \sin A , \sin B$
(B) $a , b , c$
(C) $a , \sin B , R$

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(D) $a , \sin A , R$

Let $ABC$ and $ABC^{\prime}$ be two non-congruent triangles with sides $AB=4$, $AC=AC^{\prime}=2\sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is

In $\triangle A B C$, it is known that $\overline { A B } = 4$ and $\overline { A C } = 6$, which is insufficient to determine the shape and size of $\triangle A B C$. However, knowing certain additional conditions (for example, knowing the length of $\overline { B C }$) would uniquely determine the shape and size of $\triangle A B C$. Select the correct options.
(1) If we additionally know the value of $\cos A$, then $\triangle A B C$ can be uniquely determined
(2) If we additionally know the value of $\cos B$, then $\triangle A B C$ can be uniquely determined
(3) If we additionally know the value of $\cos C$, then $\triangle A B C$ can be uniquely determined
(4) If we additionally know the area of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined
(5) If we additionally know the circumradius of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined

A triangle $A B C$ is to be drawn with $A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}$ and the angle at $A$ equal to $\theta$, where $\theta$ is a certain specified angle.
Of the two possible triangles that could be drawn, the larger triangle has three times the area of the smaller one.
What is the value of $\cos \theta$ ?
A $\frac { 5 } { 7 }$
B $\frac { 151 } { 200 }$
C $\frac { 2 \sqrt { 2 } } { 5 }$
D $\frac { \sqrt { 17 } } { 5 }$
E $\quad \frac { \sqrt { 51 } } { 8 }$
F $\frac { \sqrt { 34 } } { 8 }$

In the triangle $P Q R , P R = 2 , Q R = p$ and $\angle R P Q = 30 ^ { \circ }$.
What is the set of all the values of $p$ for which this information uniquely determines the length of $P Q$ ?

A student chooses two distinct real numbers $x$ and $y$ with $0 < x < y < 1$. The student then attempts to draw a triangle $A B C$ with:
$$\begin{aligned} A B & = 1 \\ \sin A & = x \\ \sin B & = y \end{aligned}$$
Which of the following statements is/are correct?
I For some choice of $x$ and $y$, there is exactly one triangle the student could draw.
II For some choice of $x$ and $y$, there are exactly two different triangles the student could draw.
III For some choice of $x$ and $y$, there are exactly three different triangles the student could draw. (Note that congruent triangles are considered to be the same.)
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

Find the complete set of values of $x$ for which there are two non-congruent triangles with the side lengths and angle as shown in the diagram.

A triangle $X Y Z$ is called fun if it has the following properties:
$$\begin{aligned} & \text { angle } Y X Z = 30 ^ { \circ } \\ & X Y = \sqrt { 3 } a \\ & Y Z = a \end{aligned}$$
where $a$ is a constant.
For a given value of $a$, there are two distinct fun triangles $S$ and $T$, where the area of $S$ is greater than the area of $T$.
Find the ratio
area of $S$ : area of $T$