Question

A rectangle ABCD is inscribed in a circle with centre O.Its diagonal CA is produced to a point E, outside the circle. ED is a tangent to the circle at D. If AC = 2BC,then what is the measure of DEC ?

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Use Properties of Rectangle

Step-by-step explanation:

Properties of rectangle say that all sides are parallel and angles are 90

As per theorem,

Line drawn from center of circle to the tangent of circle,  is perpendicular to it.

Hence, OD⊥ EF (which means $\angle ODF = \angle ODE = 90$°)

As in Figure,

Let us assume, $\angle CAB = \theta$

So, $sin \theta = \frac {x}{2x} = \frac {1}{2}$ = sin 30

So, $\theta = 30$°

And, $\angle CAD = 90 - 30 = 60$

Now, EAC is straight line, $\angle DAE = 180 - \angle CAD$

= 180 - 60 = 120

If a rectangle inscribed in a circle, then the diagonals are intersecting at the center of circle. i.e. point O

If $\angle DCA = 30$

$\angle DOA = 60$

In ADO, $\angle ADO = 180 - (60 + 60)$

= 60

So, $\angle ADE$ = 90 - 60 = 30 (as $\angle ODF = 90$)

And in EDO, $\angle DEO$ = 180 - (120 + 30)

= 180 - 150 = 30