Youtor
2020 AMC 10A Prob 20
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=gIK1acHHDQk
🔴 IMPORTANT RULE CHANGE FOR 2021: • *Effective 2020-2021 competition cycle, only blank scratch papers, rulers, compasses, and erasers will be permitted on the AMC 10/12, AIME, and USA(J)MO. • GRAPH PAPER AND PROTRACTOR ARE NOT ALLOWED! • 📺 Scale Drawing without Graph Paper • • Scale Drawing without Graph Paper • Tips to create and measure drawings without graph paper or protractor • 🔷 Mαth Sσlution: • ▲FEC ~ ▲AEB because they are formed from two intersecting chords inside the same circle and have: • Length Ratio, k = (5√5)/5 = √5 • Area Ratio = k² = 5 • Area[▲FEC] = ½•15•10 = 75 • Area[▲AEB] = 75/k = 75/5 = 15 • Area[▲ABC] = 4•Area[▲AEB] = 4•15 = 60 • 🤔 A good scale drawing helps spot ▲FEC ~ ▲AEB along with the convenient slopes of lines DE and FC. Also offers a good Plan B if you can't spot them! Coord Geometry and Similar Right Triangles are also good approaches, details below. • 🔶 Choose a scale factor: • I used 2/5 but a scale factor of ½ also works well, and if you're using finer graph paper, then a 1x scale might be better. You just want a good sized drawing with most of the critical points on grid. • 💜 Auxiliary Lines and Scale Drawing are two good topics to review in the week leading up to the AMC exam. Good luck on the upcoming exam! • 📺 Auxiliary Lines: • • AMC10/12 Prep: Auxiliary Lines • 📺 Scale Drawing: • • AMC10/12 Prep: Scale Drawing • ■ Coordinate Geometry Details • Place origin at the circle center: • x² + y² = 10² • Line FB has slope=½, y-int=5: • y = ½x + 5 • Substitute into circle equation: • x² + (½x + 5)² = 100 • And solve for altitude x: • 5x²/4 + 5x - 75 = 0 • x² + 4x - 60 = 0 • (x + 10)(x - 6) = 0 • Altitude, x = 6 • Area[▲ABC] = ½•20•6 = 60 • ■ Similar Right Triangle Details • Draw BG, the altitude from B to AC • Label x = EG → AG = (5 - x) • ▲DCE ~ ▲BGE → BG = 2x • BG² = AG • GC • (2x)² = (5 - x)(15 + x) • 4x² = 75 - 10x - x² • x² + 2x - 15 = 0 • (x - 3)(x + 5) = 0 → x = 3 • Altitude, BG = 2x = 6 • Area[▲ABC] = ½•20•6 = 60
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