MathDB

23

Part of 2002 AMC 12/AHSME

Problems(3)

Perpendicular Bisectors and Angle Bisectors

Source: AMC 12 A 2002 #23

8/1/2007
In triangle ABC ABC, side AC AC and the perpendicular bisector of BC BC meet in point D D, and BD BD bisects ∠ABC \angle ABC. If AD \equal{} 9 and DC \equal{} 7, what is the area of triangle ABD ABD? <spanclass=′latex−bold′>(A)</span> 14<spanclass=′latex−bold′>(B)</span> 21<spanclass=′latex−bold′>(C)</span> 28<spanclass=′latex−bold′>(D)</span> 145<spanclass=′latex−bold′>(E)</span> 285 <span class='latex-bold'>(A)</span>\ 14 \qquad <span class='latex-bold'>(B)</span>\ 21 \qquad <span class='latex-bold'>(C)</span>\ 28 \qquad <span class='latex-bold'>(D)</span>\ 14\sqrt5 \qquad <span class='latex-bold'>(E)</span>\ 28\sqrt5
geometrytrigonometryperpendicular bisectortrig identitiesLaw of Sinesangle bisectorarea of a triangle
Equal Median and Side

Source:

1/2/2009
In △ABC \triangle{ABC}, we have AB\equal{}1 and AC\equal{}2. Side BC BC and the median from A A to BC BC have the same length. What is BC BC? (A)\ \frac{1\plus{}\sqrt2}{2} \qquad (B)\ \frac{1\plus{}\sqrt3}{2} \qquad (C)\ \sqrt2 \qquad (D)\ \frac{3}{2} \qquad (E)\ \sqrt3
geometryparallelogramtrigonometry
Contrived Equation in Complex Numbers

Source:

4/6/2013
The equation z(z+i)(z+3i)=2002iz(z+i)(z+3i)=2002i has a zero of the form a+bia+bi, where aa and bb are positive real numbers. Find aa.
<spanclass=′latex−bold′>(A)</span>118<spanclass=′latex−bold′>(B)</span>210<spanclass=′latex−bold′>(C)</span>2210<spanclass=′latex−bold′>(D)</span>2002<spanclass=′latex−bold′>(E)</span>1002<span class='latex-bold'>(A) </span>\sqrt{118}\qquad<span class='latex-bold'>(B) </span>\sqrt{210}\qquad<span class='latex-bold'>(C) </span>2\sqrt{210}\qquad<span class='latex-bold'>(D) </span>\sqrt{2002}\qquad<span class='latex-bold'>(E) </span>100\sqrt2
quadraticscomplex numbersalgebraquadratic formulapolynomialRational Root Theorem