in a quadrilateral ABCD angleA=angleC=90 AE=5cm,BE=12cm and AC=21cm,DF=x cm then x=
Answers
Given : in a quadrilateral ABCD angleA=angleC=90 AE=5cm,BE=12cm and AC=21cm DF = x
To Find : x =
Solution:
∠A = ∠C = 90°
∠A + ∠C = 180° ( cyclic Quadrilateral)
Hence A , B , C & D lies on circle
AB = √5² + 12² = 13
BC = √12² +(21-5)² = 20
ΔBDA & ΔBCE
∠BDA = ∠BCE ( ∠BDA = ∠BCA angle by same chord BA and ∠BCA =∠BCE )
∠BAD = ∠BEC = 90°
=> ΔBDA ≈ ΔBCE
=> BD/BC = DA/CE = AB/BE
=> BD/20 = DA/16 = 13/12
=> BD = 65/3
AD = 52/3
BD² - BC² = CD²
=> (65/3)² - (20)² = CD²
=> (25/3)² = CD²
=> CD = 25/3
Area = (1/2) * AC * (BE + DF) = (1/2) (21) ( 12 +x)
Area = (1/2) (AB * AD + BC * CD) = (1/2) (13 * 52/3 + 20 * 25/3)
=> (1/2) (21) ( 12 +x) = (1/2) (13 * 52/3 + 20 * 25/3)
=> 63 ( 12 + x) = ( 13 * 52 + 20 * 25)
=> 63 ( 12 + x) = 1176
=> 3(12 + x) = 56
=> 12 + x = 56/3
=> x = 56/3 - 12
=> x = 20/3
=> x = 6 2/3 cm
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