Circle knowledge short test 1

Geometry Level 3

There is a circle of area $361 \pi$ (point $O$ is centre of the cirlce). Points $A$ and $B$ are located on the circle's circumference and $\measuredangle AOB$ is $\frac { 2 }{ 3 } \pi$ . A bisector of $\measuredangle AOB$ and segment $AB$ have common point $T$ . Find the value of $OT$ .

The answer is 9.5.

1 solution

Milan Milanic
Jan 9, 2016

According to formula for area of circle ( $r^{2} \pi$ ) the radius is $19$ .

$\measuredangle AOB = 120$ degrees. Therefore, angles at $A$ and $B$ are both $30$ degrees. Bisector splits that angle into two equal ones (both $60$ degrees). Then we have triangle $AOC$ or $BOC$ ( $C$ shall be common point of bisector and segment $AB$ ), both are a half of equilateral triangle. Further calculation gives that $OC = \frac{r}{2}$ .

And that is 9.5

Another really interesting question. This one was really fun, thanks!

Drex Beckman - 5 years, 5 months ago

Thanks @Drex Beckman . I didn't considered this problem that "problematic". I even thought that it was very easy and that people looked away when they saw it, thinking "This is way too easy, I don't have time for this nuisance". So, this was an encouragement for me. Thank you :)

Milan Milanic - 5 years, 5 months ago

Yeah, it's another one of your well thought out ones. I really appreciate your originality, and the problems are really fun to solve. Started following you, by the way. Thanks.

Drex Beckman - 5 years, 5 months ago

Circle knowledge short test 1

The answer is 9.5.

1 solution

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