Bidyut Guha Thakurta and Amitava Mitra Solutions for Exercise 1: EXERCISE 6

Two sides adjoining the right angle of a right-angled triangle are $a$ and $b$. If the length of the hypotenuse is $\sqrt{a^x+b^x}$, then the value of $x=$ _____.

In a triangle $RST$, $R{S}^2+S{T}^2=R{T}^2$, then the angle _____ is a right angle.

If two adjacent sides of a rectangle are $8 \mathrm{cm}$ and $6 \mathrm{cm}$, then the length of a diagonal is _____ $\mathrm{cm}$.

$ABC$ is a right-angled isosceles triangle of which $\angle C$ is a right angle. Then $A{B}^2=2B{C}^2$.

In a triangle $PQR$, if $P{Q}^2+Q{R}^2=P{R}^2$, then $\angle P$ is a right angle.

$AC$ and $BD$ are two diagonals of a rhombus $ABCD$. Then $A{C}^2+B{D}^2=2A{B}^2$

In an isosceles right-angled triangle if the length of each of the two equal sides is $2\sqrt{2} cm$, then the length of the hypotenuse is $4 cm$.

$ABC$ is an equilateral triangle and $AD\perp BC$. Then $4A{D}^2=3A{B}^2$.