Question

in fig 1 determine,is ∆ ABC is right angled at B?

Answer 1

if <B =90
according to Pythagoras theorem
AB²+BC²= AC²
from the fig 1
AB= 2,BC=2,AC=3
2²+2²=3²
4+4=9
8=9 [false]
therefore
<B ≠90
 

Answer 2

Q1 
Take the biggest side = AC = 3cm
check if  AC² = AB² + BC² 
             9     =  2² + 2²
       
Not true.  Hence Pythagoras theorem is not applicable for ΔABC. So it is not a right angled triangle.

Q2:
        Angle at R is same in both triangles.  (vertical angles/opposite angles). There is a right angle in each triangle.   As the sum of three angles is 180 deg, the third angles are equal in both triangles.
        So they are similar triangles.

Q3:
   in ΔDNE:   DN² = DE² + EN² = (9+4)² = 169

   in ΔNEQ:   EQ² = NE² - 9²
   in ΔDEQ :   EQ² = DE² - 4²
 Add these two equations:
   EQ² = [NE² + DE² - 81 - 16]/2
        = (169-81-16)/2                  

EQ = 6