Properties of Triangles Questions with Solutions

In $△ABC$, sides are $a=9$, $b=7$, $c=12$. Find angle $C$ (opposite side $c$) using the Law of Cosines.

The perimeter of a triangle ABC is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

Let $\Delta ABC$ be a triangle whose area is $10\sqrt{3}$ units with side lengths $\left|AB\right|=8$ units and $\left|AC\right|=5$ units. Find possible values of the angle A.

In a $\Delta ABC$, $\left(c+a+b\right)\left(a+b-c\right)=ab$. The measure of the angle C is

If the sides of a triangle are $x^2+x+1$, $2x+1$ and $x^2-1$, then its largest angle is

In $△ABC$, $R$ is the circumradius and $8R^2=a^2+b^2+c^2$. Then $△ABC$ is

In a $△ABC$, if ${tan}^2\frac{A}{2}+{tan}^2\frac{B}{2}+{tan}^2\frac{C}{2}=k$, then $k$ is always

In a triangle, if the sum of two sides is $x$ and their product is $y$ such that $\left(x+z\right)\left(x-z\right)=y$, where $z$ is the third side of the triangle, then the triangle is

In a triangle ABC, if $a{cos}^2\frac{C}{2}+c{cos}^2\frac{A}{2}=\frac{3b}{2}$, then the sides of the triangle are in

If A, B and C are three angles of a $△ABC$, whose area is $\Delta$. Let a, b and c be the sides opposite to the angles A, B and C respectively. If $s=\frac{a+b+c}{2}=6$, then the product $\frac{1}{3}{s}^2\left(s-a\right)\left(s-b\right)\left(s-c\right)$ is equal to

If the angles of a triangle are in the ratio 2 : 3 : 7, then the ratio of the sides opposite to these angles is

If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in HP, then sin A, sin B, sin C are in

In a triangle ABC, let $\angle C=\frac{\pi }{2}$. If r is the inradius and R is the circumradius of the triangle ABC, then $2\left(r+R\right)$ equals

In an acute-angled triangle ABC, the least value of $secA+secB+secC$ is

If $\Delta =a^2-{\left(b-c\right)}^2$, where $\Delta$ is the area of the triangle ABC, then $tanA=$