Angles problem?

2 Answers
Apr 22, 2018

See below.

Explanation:

First note.

The tangent to a point on the circumference of a circle is always perpendicular to the radius at the same point.

From diagram:

#/_OBQ=90^@#

Hence:

#/_CBO=90^@-2X#

For triangle #COB#

#bb(CO)=bb(BO)# radii of circle.

So:

#COB# is isosceles.

Hence:

#/_OCB=/_CBO=90^@-2X#

For triangle #DOC#

#bb(CO)=bb(DO)# radii of circle.

#DOC# is isosceles.

Hence:

#/_DCO=/_CDO=X#

For #/_BCD#

We know:

#/_OCB=90^@-2X#

and

#/_DCO=X#

Hence:

#/_BCD=/_OCB+/_DCO=90^@-2X+X=90^@-X#

#/_COB=180^@-(/_CBO+/_OCB)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(90^@-2X +90^@-2X)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(180^@-4X)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ =4X#

#/_DOC=180^@-(/_CDO+/_DCO)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(X+X)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-(2X)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ =180^@-2X#

#/_DOB+/_DOC+/_COB=360^@#

#:.#

#/_DOB=360^@-/_DOC-/_COB#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \=360^@-(180^@-2X)-4X#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \=360^@-180^@+2X-4X#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \=180^@-2X#

#/_ODA = /_OBA=90^@# Perpendicular.

#/_ODA+/_OBA+/_DOB+y=360^@# Sum of angles in quadrilateral

#:.#

#90^@+90^@+(180^@-2X)+y=360^@#

#y=360^@-180^@-180^@+2X#

#y=2X#

Collecting everything:

#/_DCO=X#

#/_CBO=90^@-2X#

#/_OCB=90^@-2X#

#/_BCD=90^@-X#

#/_DOB=180^@-2X#

#y=2X#

I think there is probably a simpler method for this and maybe someone else will give a more concise way, but hope it helps.

Apr 22, 2018

see explanation.

Explanation:

enter image source here
let #r# be the radius of the circle,
As #OD=OC=r, => angleDCO=angleCDO=x#,
given #QBA# is tangent to the circle and #angleCBQ=2x#,
#=> angleCBO=90-2x#,
as #OB=OC=r, => angleOCB=angleOBC=90-2x#,
#angleBCD=angleOCB+angleDCO=90-2x+x=90-x#,
As the angle at the center of the circle is twice the angle at the circumference
#angleDOB=2*angleBCD=2*(90-x)=180-2x#

As #ADOB# is a quadrilateral, and the sum of the angles of a quadrilateral always adds up to #180^@#,
#=> y+angleADO+angleDOB+angleABO=360^@#,
#=> y=360-90-(180-2x)-90=360-360+2x=2x#