Difference between revisions of "2024 SSMO Accuracy Round Problems/Problem 5"
(Created page with "==Problem== Let <math>ABCD</math> be a convex quadrilateral such that <math>\angle ABC = 120^\circ</math> and <math>\angle ADC = 60^\circ</math>. If <math>AB = BC = CD = 5</m...") |
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+ | From the Law of Cosines on triangle <math>ABC</math>, we have <cmath>AC^2 = 5^2+5^2-2\left(-\frac{1}{2}\right)\cdot5\cdot5\implies AC = 5\sqrt{3}.</cmath> Let <math>AD = x.</math> From the Law of Cosines on triangle <math>ADC,</math> we have <cmath>x^2+5^2-2 \left(\frac12\right)5\cdot x = \left(5\sqrt{3}\right)^2\implies x^2-5x-50 = 0\implies x = 10.</cmath> Finally, we use the sine area formula: <cmath>[ABCD] = [ABC]+[ACD] = \frac{1}{2}\left(5\cdot5\cdot\left(\frac{\sqrt{3}}{2}\right)+10\cdot5\left(\frac{\sqrt{3}}{2}\right)\right) = \frac{75\sqrt{3}}{4}\implies 75+3+4 = \boxed{82}.</cmath> | ||
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+ | ~SMO_Team |
Latest revision as of 14:31, 10 September 2025
Problem
Let be a convex quadrilateral such that
and
. If
, the area of
can be expressed as
where \(a,b,\) and \(c\) are positive integers and \(c\) is squarefree. Find
Solution
From the Law of Cosines on triangle , we have
Let
From the Law of Cosines on triangle
we have
Finally, we use the sine area formula:
~SMO_Team