If the lines shown in the picture touch the circle with center O only

Question

https://gmatclub.com/forum/download/file.php?id=51278 If the lines shown in the picture touch the circle with center O only at points A and B, what is the area of the triangle ABC? (1) Triangle ADC has area of 25. (2) Triangle BDC has area of 25. image008.gif

Kinshook · Answer

If the lines shown in the picture touch the circle with center O only at points A and B, what is the area of the triangle ABC?
Since the figure is symmetrical w.r.t to line CDO, angle ADC = angle BDC = 90
Area ADC = CD * DA
Area BDC = CD * DB
Since DA = DB; and DA + DB = AB; DA=DB= 1/2*AB; Area ADC = Area BDC = 1/2 * Area ABC

(1) Triangle ADC has area of 25.
Area of triangle ADC = 1/2 * Area of triangle ABC = 25
Area of triangle ABC = 25*2 = 50
SUFFICIENT

(2) Triangle BDC has area of 25.
Area of triangle BDC = 1/2 * Area of triangle ABC = 25
Area of triangle ABC = 25*2 = 50
SUFFICIENT

IMO D

mafuz · Answer

Hi! how D creates two right angle. Can anyone explain. I did not understand above explanation as it said ...figure is symmetrical to CDO..

Aviral1995 · Answer

mafuz

This can be said by applying the properties of a triangle
Consider triangle ACO and BCO
Here AO=OB(BOTH ARE RADIUS OF CIRCLE WITH CENTRE O); Angle CAO and Angle CBO are 90 degree; CA=CB(both are tangent)

So we can say by property SAS that triangle CAO is congruent to Triangle CBO

Now, consider triangle ADO and Triangle BDO
Here, AO=BO(Radius); angle AOD = angle BOD(using congruent condition of traingle ACO and BCO); OD=OD(Same side)
So, traingle ADO and BDO are congruent

Now since ADC is a chord, and angle ADO and BDO are same; we can say they are 90 degree

Hope this helps!

mafuz · Answer

Thanks a lot  aviral

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If the lines shown in the picture touch the circle with center O only

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