Properties of Triangles

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 $45^{\circ}$…

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: 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,…

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?

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 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

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

$A B C$ is a triangle in which $B C=10 \mathrm{~cm}, C A=6 \mathrm{~cm}$ and $A B=8 \mathrm{~cm}$. Which one of the following is correct?

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?

Consider a triangle ABC satisfying $2 \mathrm{a} \sin ^2\left(\frac{\mathrm{C}}{2}\right)+2 \mathrm{c} \sin ^2\left(\frac{\mathrm{~A}}{2}\right)=2 \mathrm{a}+2 \mathrm{c}-3 \mathrm{~b}$ $\sin \mathrm{A}, \sin…

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?

The angles A, B and C of a triangle ABC are in the ratio 3 : 5 : 4. What is the value of $a+b+\sqrt{2} c$ equal to?