Setting Sail

Geometry Level 3

Angle $ABC=90^\circ$ , angle $CAD=45^\circ$

Which area is larger, green or yellow?

Brown circle is centered at $B$ , yellow circle at $A$ .

Inspired by Find This Area by Victor Paes Plinio

Green has a larger area Yellow has a larger area They areas are the same

1 solution

Marta Reece
Jul 7, 2017

Let’s set $\overline{AB}=\overline{BC}=1$

The combined area of green and brown is a quarter circle and it is equal to $\dfrac{\pi}4$

The hypotenuse of $\triangle ABC=\sqrt{1+1}=\sqrt2$

The combined area of brown and yellow is an eighth of a circle and it is equal to

$\dfrac{2\pi}8=\dfrac{\pi}4$

Since

${\color{#CEBB00}{YELLOW}}+{\color{#624F41}BROWN}={\color{#20A900}GREEN}+{\color{#624F41}BROWN}$

It is also true that

${\color{#CEBB00}{YELLOW}}={\color{#20A900}GREEN}$

Could not solve the problem because not enough data. In the solution someone assume AB = BC which was not given in original problem.

Mike Raguin - 3 years, 11 months ago

This comes from the fact that the brown circle, centered at B, goes from A to C.

Marta Reece - 3 years, 11 months ago