Question

A trapezoid has side lengths $3,$ $5,$ $7,$ and $11.$ The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1} + r_2\sqrt{n_2} + r_3,$ where $r_1,$ $r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to $r_1 + r_2 + r_3 + n_1 + n_2?$

$57$

$59$

$61$

$63$

$65$

Solution:

For a trapezoid with parallel sides $a\lt c$ and legs $b,d,$ translating a leg forms a triangle with sides $b,$ $d,$ and $c-a.$ The triangle inequality forces the longer parallel side to be $c=11.$

If $a=3,$ the triangle has sides $5,7,8$ with area $10\sqrt3,$ and the trapezoid has area $\tfrac{35}{2}\sqrt3.$ If $a=5,$ the triangle has sides $3,6,7$ with area $4\sqrt5,$ giving trapezoid area $\tfrac{32}{3}\sqrt5.$ If $a=7,$ the triangle has sides $3,4,5,$ a right triangle, giving trapezoid area $27.$

The total is $\tfrac{35}{2}\sqrt3+\tfrac{32}{3}\sqrt5+27,$ so $r_1+r_2+r_3+n_1+n_2=\tfrac{35}{2}+\tfrac{32}{3}+27+3+5=63+\tfrac16.$

The greatest integer at most this value is $63.$

Thus, the correct answer is D.

2012 AMC 12B Problem 20

Solution:

Problem 20 in Other Years