Triangle and Circles Basics

Level 2

$\Delta ABC$ is a right angled triangle with $AB = 3 cm$ . A circle is touching its sides $AB$ and $AC$ with centre $O$ and diameter $BD$ . $BD = 2 cm$ . Length of $CD$ is $\dfrac{m}{n}$ Give your answer as $m^3-m^2+n^3-n^2$ . $m,n$ are co-prime

The answer is 48.

1 solution

Marta Reece
Dec 21, 2017

$tan(\angle BAO)=\frac13$

$\angle EOC=\angle BAC$

$\angle BAC=2\times \angle BAO$

$EC=tan(\angle EOC)=tan(2\times \angle BAO)=\frac{2\times tan(\angle BAO)}{1-tan^2(\angle BAO)}=\frac{\frac23}{1-\frac19}=\frac34$

$OC^2=OE^2+EC^2$

$OC^2=1+\frac9{16}=\frac{25}{16}$

$DC=OC-1=\frac54-1=\frac14$

$m=1, n=4$

Answer $=1-1+64-16=\boxed{48}$

I really like the problem. I think it's elegant.

I feel that asking the solver to report the answer in a complicated form does not contribute to the overall effect, though.

Marta Reece - 3 years, 5 months ago

The problem is elegant if solved without any trigonometry.What are your views??

Sumukh Bansal - 3 years, 5 months ago

The answer is to be answered in a complicated form because if it is not so a person might get a trial-and-error correct answer.

Sumukh Bansal - 3 years, 5 months ago

Triangle and Circles Basics

The answer is 48.

1 solution

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