Question

let ABCD be a quadilateral inscribd in a circle with diameter AC ,E be the foot of perpendicular from D onto AB as shown in the figure if AD=DC and the area of ABCD is 24 cm² .find the square length of DE(in cm)​

Answer 1

ANSWER :

Let CD= x then AB = 2CD = 2x

Let r be the radius of the circle inscribed in the quadrilateral ABCD.

ar ( square ABCD ) = 18 and AB||CD

=> 1/2 (x+2x)2r = 18

=>3xr = 18

=> xr = 6. ------------------(1)

OP = OM = PD = OQ = AM = r

=> PC = x-r and MB = 2x-r

Let angle PCQ = angle OCQ = O

In right - angled ΔOPC,

tanΦ = OP/CP = r/x-r. -------------(2)

CD||AB

.°. angle PCB = angle QOM = 20

=> angle CBA = 180°- 20

=> angle OBM = 90° - 0

From ΔOMB

tan (90° - Φ) = OM/MB = r/2x-r

Right - angled ΔOBM,

tan Φ = 2x-r/r. --------------------(3)

From (2) and (3)

=> r/x-r 2x-r/r

=> 2x² - 3xr = 0

=> x(2x-3r) = 0

=> x=3r/2. ---------------------(4)

From (1) and (4)

=> xr = 6

=> 3xr/2 = 6

=> r² = 4

=> r= 2