Problem #1519


Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(a,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $a$? [asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt));  fill((2/3,0)--(3,3)--(3,1)--(2,1)--(2,0)--cycle,gray);  xaxis("$x$",-0.5,4,EndArrow(HookHead,4)); yaxis("$y$",-0.5,4,EndArrow(4));  draw((0,1)--(3,1)--(3,3)--(2,3)--(2,0)); draw((1,0)--(1,2)--(3,2)); draw((2/3,0)--(3,3));  label("$(a,0)$",(2/3,0),S); label("$(3,3)$",(3,3),NE); [/asy]

$\text{(A) } \frac 12 \qquad \text{(B) } \frac 35 \qquad \text{(C) } \frac 23 \qquad \text{(D) } \frac 34 \qquad \text{(E) } \frac 45$

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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