Question

Find the length of the diagonal of the
rectangle BCDE, if angleBCA=angleDCF .AC=6m.
and CF= 12cm

Answer 1

Given
BCDE may be a parallelogram, them
∠BCD = ∠CDE = ∠DEB = EBC = 90°
C, ∠ACD and ∠DCF from a linear pair then
=> ∠ACD  + ∠DCF =180°
=> ∠BCA +∠BCD+∠DCF= 180°
=> ∠BCA+90° +∠DCF =180°
=> ∠BCA +∠DCF= 90°
=> ∠BCA +∠ BCA=90° [∴∠BCA =∠DCF]
=> 2∠BCA =90°
=> ∠BCA = 90°/2 = 45°
ΔBAC we have
cos ∠BCA =
=> cos 45° = AC/BC
=>
=> BC =6cm
=>Δ DFC =  
=> cos 45° = CF/CD
=>
Let BD be the diagonal of rectangular BCDE
ΔBCD we have
=> BD = BC + CD
(By phytagoras theorem)
=> BD= (6)+(12)
=> 72+288 = 360
=> BD = 360
=> BD  = 6 cm

Hope it helps