MathDB

12

Part of 2020 AMC 12/AHSME

Problems(2)

Line Rotation

Source: 2020 AMC 12A #12

1/31/2020
Line ℓ\ell in the coordinate plane has the equation 3x−5y+40=03x - 5y + 40 = 0. This line is rotated 45∘45^{\circ} counterclockwise about the point (20,20)(20, 20) to obtain line kk. What is the xx-coordinate of the xx-intercept of line k?k?
<spanclass=′latex−bold′>(A)</span>10<spanclass=′latex−bold′>(B)</span>15<spanclass=′latex−bold′>(C)</span>20<spanclass=′latex−bold′>(D)</span>25<spanclass=′latex−bold′>(E)</span>30<span class='latex-bold'>(A) </span> 10 \qquad <span class='latex-bold'>(B) </span> 15 \qquad <span class='latex-bold'>(C) </span> 20 \qquad <span class='latex-bold'>(D) </span> 25 \qquad <span class='latex-bold'>(E) </span> 30
AMC 12AMC 12 Aanalytic geometryrotationgeometry2020 AMC 12A2020 AMC
Two Chords and a 45 Degree Angle

Source: 2020 AMC 12B #12

2/6/2020
Let AB‾\overline{AB} be a diameter in a circle of radius 52.5\sqrt2. Let CD‾\overline{CD} be a chord in the circle that intersects AB‾\overline{AB} at a point EE such that BE=25BE=2\sqrt5 and ∠AEC=45∘.\angle AEC = 45^{\circ}. What is CE2+DE2?CE^2+DE^2?
<spanclass=′latex−bold′>(A)</span> 96<spanclass=′latex−bold′>(B)</span> 98<spanclass=′latex−bold′>(C)</span> 445<spanclass=′latex−bold′>(D)</span> 702<spanclass=′latex−bold′>(E)</span> 100<span class='latex-bold'>(A)</span>\ 96 \qquad<span class='latex-bold'>(B)</span>\ 98 \qquad<span class='latex-bold'>(C)</span>\ 44\sqrt5 \qquad<span class='latex-bold'>(D)</span>\ 70\sqrt2 \qquad<span class='latex-bold'>(E)</span>\ 100
AMC 12 BAMC 12AMC2020 AMC2020 AMC 12Bgeometry