Properties of Triangles

Paragraph: In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the in-circle of the triangle touches the sides $P Q, Q R$ and $R P$ at $N, L$ and $M$ respectively such that the lengths $P N,…

In a triangle $A B C, a=(1+\sqrt{3}) \mathrm{cm}, b=2 \mathrm{~cm}$ and angle $C=60^{\circ}$. Then the other two angles are

ABC is a triangle inscribed in a circle with centre O . Let $\alpha=\angle \mathrm{BAC}$, where $45^{\circ} \lt \alpha \lt 90^{\circ}$. Let $\beta=\angle \mathrm{BOC}$. Which one of the following is correct?

Consider the following statements : If $A B C$ is a right-angled triangle, right-angled at $A$ and if $\sin B=\frac{1}{3}$, then $\operatorname{cosec} C=3$. If $b \cos B=c \cos C$ and if the triangle $A B C$ is not…

Paragraph: $A B C$ is a triangular plot with $A B=16 \mathrm{~m}, B C=10 \mathrm{~m}$ and $C A=10 \mathrm{~m}$. A lamp post is situated at the middle point of the side $A B$. The lamp post subtends an angle $45^{\circ}$…

The sides of a triangle are $m, n$ and $\sqrt{m^2+n^2+m n}$. What is the sum of the acute angles of the triangle?

In triangle $\mathrm{ABC}, \mathrm{BC}=\sqrt{39}, \mathrm{AC}=5$ and $\mathrm{AB}=7$. What is the measure of the angle $A$ ?

Let $a, b, c$ be the lengths of sides $B C, C A, A B$ respectively of a triangle $A B C$. If $p$ is the perimeter and $q$ is the area of the triangle, then what is $p(p-2 a) \tan \left(\frac{A}{2}\right)$ equal to?

Paragraph: In the triangle $A B C, a^2+b^2+c^2=a c+\sqrt{3} b c$ Question: What is the nature of the triangle?

If in a triangle $A B C, \sin ^3 A+\sin ^3 B+\sin ^3 C$ $=3 \sin \mathrm{~A} \sin \mathrm{~B} \sin \mathrm{C}$, then what is the value of the determinant $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a &…

The sides $a, b, c$ of a triangle ABC are in arithmetic progression and ' $a$ ' is the smallest side. What is $\cos \mathrm{A}$ equal to?

$A B C$ is a triangle right angled at $B$. The hypotenuse $(A C)$ is four times the perpendicular $(B D)$ is four times the perpendicular $(B D)$ drawn to its from the opposite vertex and $A D \lt D C$. What is one of…