2024 SSMO Speed Round Problems/Problem 5

Problem

Let $\triangle ABC$ and $\triangle ADC$ be right triangles, such that $\angle ABC = \angle ADC = 90^\circ$ . Given that $\angle ACB = 30^\circ$ and $BC = 3\sqrt{3}$ , find the maximum possible length of $BD$ .

Solution

Note that quadrilateral $ABCD$ is a cyclic quadrilateral with diameter $AC.$ Since $\angle{ACB} = 30^\circ$ and $BC = 3\sqrt{3},$ we have $AC = \frac{BC}{\cos\angle{ACB}} = \frac{3\sqrt{3}}{\frac{\sqrt{3}}{2}} = 6.$ So, the maximum possible length of $BD$ is $6,$ as the maximum length of any chord on a circle is the diameter.

~SMO_Team

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2024 SSMO Speed Round Problems/Problem 5

Problem

Solution