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What is the greatest possible area of a triangular region

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What is the greatest possible area of a triangular region

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What is the greatest possible area of a triangular region with one vertex at the center of a circle with radius one and the other two vertices on the circle?
$$A.\ \frac{ \sqrt{3}}{4}$$
$$B.\ \frac{1}{2}$$
$$C.\ \frac{\pi}{4}$$
$$D.\ 1$$
$$E.\ \sqrt{2}$$
OA B

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AAPL wrote:
GMAT Prep

What is the greatest possible area of a triangular region with one vertex at the center of a circle with radius one and the other two vertices on the circle?
$$A.\ \frac{ \sqrt{3}}{4} \,\,\,\,\,\,B.\ \frac{1}{2}\,\,\,\,\,\,C.\ \frac{\pi}{4}\,\,\,\,\,\,D.\ 1\,\,\,\,\,\,E.\ \sqrt{2}$$


$$?\,\,\, = \,\,{S_{\Delta ABC}}\,\,\max $$

Let C be the center of the circle with unitary radius.

Without loss of generality, we may (and will) assume point A is one unit at the right of point C (as shown in the figure on the left).

For the last vertex (B), without loss of generality (in terms of exploring possible areas) there are only two possibilities:


(1) B is in the arc AD (figure in the middle) or (2) B is in the arc DE (figure on the right)


In BOTH cases we have:



$${S_{\Delta ABC}} = {{AC \cdot h} \over 2} = {h \over 2}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = {1 \over 2}\,\,\,\,\,\,\left( {{\rm{when}}\,\,h = CD\,,\,\,{\rm{i}}{\rm{.e}}{\rm{.}},\,\,B = D} \right)$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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GMAT/MBA Expert

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AAPL wrote:
GMAT Prep

What is the greatest possible area of a triangular region with one vertex at the center of a circle with radius one and the other two vertices on the circle?
$$A.\ \frac{ \sqrt{3}}{4}$$
$$B.\ \frac{1}{2}$$
$$C.\ \frac{\pi}{4}$$
$$D.\ 1$$
$$E.\ \sqrt{2}$$
OA B
Given two sides of a triangle, the greatest possible area will be achieved if the two sides form a RIGHT ANGLE and serve as the triangle's base and height.
For the problem above, I posted an illustrated solution here:
https://www.beatthegmat.com/largest-possible-area-of-triangle-inside-circle-t274809.html

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Mitch Hunt
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GMATGuruNY@gmail.com

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.
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