# Math Competition Question Hong Kong 2

Note: This is from the Po Leung Kuk 2012 paper. The previous one I posted was from the Po Leung Kuk 2011.
Here's another question for everyone:
There is a figure made out of two rows of identical squares: three squares in the top row and four in the bottom, so that the first three squares in the bottom row form a $2\times 3$ rectangular block of squares with the squares of the first row. (If anyone knows how to insert a diagram, please do so.) A line, $l$, passes through the top side of the second square on the top row at point $E$ and passes through the bottom side of the third square from the bottom row at point $F$. Line $l$ cuts the figure into 2 equal pieces.
Let the top left corner of the figure be point $A$ and the bottom left be point $C$. If $AE+CF=91$, what is the area of an individual square.
I have no idea on how to approach this problem. A hint on how to start the problem is good enough. Also, if anyone knows how to create a diagram, please put it in the comments.
Thanks!

• Is the figure perfectly symmetric about the line between the 2nd and 3rd squares on the bottom row? – JimmyK4542 Jun 11 '14 at 2:44
• nope. Both rows start from the same column. – user148697 Jun 11 '14 at 2:46
• @jonnytan999 I guess the hardest part of this question is -- (1) guessing what the figure looks like from your description; and (2) The meaning of your 'EQUAL' [are the two equal pieces EQUAL in AREA or EQUAL in shape! – Mick Jun 11 '14 at 15:18
• can you tell me how to insert the diagram, please... – user148697 Jun 12 '14 at 5:03
Let $x$ be the sidelength of a square. Then the area of the entire figure is $7x^2$.
One half of the figure is quadrilateral $ACFE$, which is a trapezoid with height $2x$ and bases $AE$ and $CF$.
Can you figure out the area of this trapezoid in terms of $x$? Then, you should be able to figure out what value of $x$ makes this trapezoid have an area of half the area of the figure.