Question

A circle is circumscribed about a triangle with sides 20, 21 and 29, thus dividing the interior of the circle into 4 regions. Let A, B and C be the areas of the non triangular regions, with C being the largest. Then which one of the following holds True?

• A. $A+B=C$
• B. $A+B+210=C$
• C. $A^2+B^2=C^2$
  
• D. $20A+21B=29C$

Answer 1

The correct option is B $A+B+210=C$

${20}^2+{21}^2=841={21}^2$
Therefore triangle is right angle triangle
But then its hypotenuse is a diameter of the circumcircle, and thus c is exactly one half of the circle.
Moreover, the area of the triangle is $\frac{20.21}{2}=210$
Therefore the are of the other half of the circumradius can be expressed as $A+B+210$
thus option (b) is correct
Note that option (a) is false as $A+B<C$, we have $A^2+B^2<{(A+B)}^2<C^2$ and thus option (c) is false
Similarly $20A+21B<\left(21\right(A+B)<21C<29C$ thus option (d) is false