S Purkayastha Solutions for Chapter: The Triangle and its Properties, Exercise 8: Brain Workout

In $△ABC$, if $2\angle A=3\angle B=6\angle C,$ calculate the measure of $\angle A, \angle B$ and $\angle C$.

$($ Hint: Let $2\angle A=3\angle B=6\angle C=x\Rightarrow \angle A=\frac{x}{2}, \angle B=\frac{x}{3},\angle C=\frac{x}{6}$$)$

In the following figure, the sides $AB$ and $AC$ of $△ABC$ are produced to $P$ and $Q$, respectively. The bisectors of $\angle PBC$ and $\angle QCB$ intersect at $O$. Prove that $\angle BOC=90°-\frac{1}{2}\angle BAC$.

The diagonals of a rhombus measure $16 \mathrm{cm}$ and $30 \mathrm{cm}$. Find its perimeter.

$△ABC$ is an isosceles triangle, right-angled at $C$. Prove that $A{B}^2=2A{C}^2$. 

The hypotenuse of a right-angled triangle is $6 \mathrm{m}$ more than twice the shortest side. If the third side is $2 \mathrm{m}$ less than the hypotenuse, find the sides of the triangle.

In the following figure, $\angle BAC=90°, \angle ADC=90°$. Find the length of $AB$.