# in a pentagon abcde, ea=ab=bc=cd=10cm, angle eab = angle ecd = 90degrees angle abc = 135 degrees what is the length of ed??????Please help me answer this question!!!!!!!

vaaruni | High School Teacher | (Level 1) Salutatorian

Posted on

Given : A pentagon abcde, ea=ab=bc=cd=10cm,                             angle eab= angle ecd=90 degrees  and angle abc=135 degrees. We require to find length of ed ?

join eb and ec. We get three triangles, triangle eab, triangle ebc and triangle aec.  Let us take the triangle eab which is a right angled triangle [ angle eab = 90 degrees (given) ]

eb^2 = ea^2 + ab^2  ( using pythagoras theorem)

Or, eb^2 = (10)^2 + (10)^2)

Or, eb^2 = 100 + 100  = 200

eb^2 = 200  ------ (1)

Triangle eab is an isosceles triangle in which  ea=ab, therefore angle aeb= angle abe =45  degrees ( angle eab=90 degrees ) . Now         since, angle abc = 90 degtees (given) and angle abe=45 degrees , Therefore angle ebc=90 degrees.Taking right angle triangle ebc :

ec^2 = eb^2 + bc^2 ( using pythagoras theorem)

Or, ec^2 = 200 + (10)^2  [ eb^2= 200, from eqn. (1) ]

Or, ec^2 = 200 + 100 = 300

ec^2 = 300  ---------(2)

Next taking right angled triangle ecd(angle ecd =90degrees,given)

ed^2 = ec^2 + dc^2 ( using pythagoras theorem )

Or,   ed^2 = 300 + (10)^2 [ ec^2=300 from equation(2) ]

Or,   ed^2 = 300 + 100 = 400

Or,   ed^2 = 400

therefore  ed = square root(400) = 20