2013 Mock AIME I Problems/Problem 5

Revision as of 10:11, 30 July 2024 by Thepowerful456 (talk | contribs) (Solution)

Problem

In quadrilateral $ABCD$, $AC\cap BD=M$. Also, $MA=6, MB=8, MC=4, MD=3$, and $BC=2CD$. The perimeter of $ABCD$ can be expressed in the form $\frac{p\sqrt{q}}{r}$ where $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime number. Find $p+q+r$.

Solution

[asy]  import geometry;  // Defining Points point O = origin; point B = (1,0); point A = dir(115.583); point C = dir(-115.583); point D = dir(-165.638); point M;  // Circle draw(circle(O, 1));  // Quadrilateral and Diagonals draw(A--B--C--D--cycle); draw(A--C); draw(B--D);  // Defining M pair[] m = intersectionpoints((A--C),(B--D)); M = m[0];  // Labelling Points dot(A); label("A",A,NW); dot(B); label("B",B,E); dot(C); label("C",C,SW); dot(D); label("D",D,WSW); dot(M); label("M",M,NE);  // Length Labels label("$3$", midpoint(D--M), NNW); label("$8$", midpoint(M--B), NNW); label("$6$", midpoint(A--M), E); label("$4$", midpoint(C--M), E); label("$2x$", midpoint(B--C), SE); label("$x$", midpoint(C--D), NE);  [/asy]

$\boxed{037}$.

See also