Find area of this circle

Question

Given the circle O with perpendicular diameters and a chord, find the area of the circle if $EF = 8"$ and $DE = 20"$ inches.

Answer 1

Hint: Draw the line segment $BE$ and use the fact that $\triangle ODF$ and $\triangle EDB$ are similar.

Answer 2

As Andre said in his answer, $\Delta ODF$ ~ $\Delta EDB$. Using the fact that similar triangle's corresponding sides are in proportion we can set up a proportion. The eqaution is: $$\frac {DF}{DO}=\frac {DB}{DE}$$ We can assign the variable $x$ to $DO$ and we know that $DB=2x$ because $DB$ is a diameter and $DO$ is a radius. Now. it's as simple as substituting 12 for $DF$, $x$ for $DO$, $2x$ for $DB$ and 20 for $DE$ in the original proportion. We get: $$\frac {12}{x}=\frac{2x}{20} \\ 2x^2=240 \\ x=\sqrt{120}$$ We now have the length of the radius, so we can use that in the formula for the area of a circle, $A=\pi r^2$, to find that the area is equal to $120\pi"$ or about 376.99". If you would like to, you can check the length of the radius using trigonometry, specifically the Law of Cosines.