The diagram shows four straight lines AD, BC, AC, and BD. lines AC BD intersect at O such that angle AOB is #pi/6radians#. AB is an arc of the circle, centre O and radius 10cm, and CD is an arc of the circle, centre O and radius 20cm?

Question

Find the perimeter of ABCD

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Answer 1 · 2018-02-24T17:02:09

#73.89color(white)(88)# 2.d.p.

From the diagram:

#/_COB=pi-pi/6=(5pi)/6#

These are angles on the straight line #bb(AC)#

#/_DOA=pi-pi/6=(5pi)/6#

These are angles on the straight line #bb(DB)#

#/_DOC=pi-(5pi)/6=pi/6#

These are angles on the straight line #bb(DB)#

Solving #Delta color(white)(88)AOD#

Using cosine rule:

#AD^2=OD^2+OA^2-2(OD)(OA)cos(O)#

#AD^2=(20)^2+(10)^2-2(20)(10)*cos((5pi)/6)#

#AD^2=400+100-400*(-sqrt(3)/2)#

#AD^2=500-400*(-sqrt(3)/2)=500+200*sqrt(3)#

#AD=sqrt(500+200*sqrt(3))=29.09color(white)(88)# 2 .d.p.

#CB=AD# Same dimensions

Arc length is: #color(white)(88)rtheta#

Length of arc #CD#

#20*pi/6=(10pi)/3#

Length of arc #AB#

#10*pi/6=(5pi)/3#

Perimeter is:

#AD +CB +(10pi)/3+(5pi)/3#

#2(sqrt(500+200*sqrt(3))) +5pi=73.89color(white)(88)# 2.d.p.