# 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!!!!!!!

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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

Hence **ed = 20cm. Answer**