MathDB

14

Part of 2005 AIME Problems

Problems(2)

Area of an Unique Square

Source:

11/13/2005
Consider the points A(0,12)A(0,12), B(10,9)B(10,9), C(8,0)C(8,0), and D(−4,7)D(-4,7). There is a unique square SS such that each of the four points is on a different side of SS. Let KK be the area of SS. Find the remainder when 10K10K is divided by 10001000.
geometryanalytic geometrygraphing linessloperotationtrigonometry
Side Length of a Triangle

Source: Aime 2005b, #14

11/14/2005
In triangle ABCABC, AB=13AB=13, BC=15BC=15, and CA=14CA=14. Point DD is on BC‾\overline{BC} with CD=6.CD=6. Point EE is on BC‾\overline{BC} such that ∠BAE≅∠CAD.\angle BAE\cong \angle CAD. Given that BE=pqBE=\frac pq where pp and qq are relatively prime positive integers, find q.q.
trigonometrygeometrycircumcircleanalytic geometrynumber theoryrelatively primetrig identities