M203 20260328 Polygon
If $a,b$, and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then
1977 AHSME Problems/Problem 30
$\textbf{(A) }d=a+b\qquad \textbf{(B) }d^2=a^2+b^2\qquad \textbf{(C) }d^2=a^2+ab+b^2\qquad\\ \textbf{(D) }b=\frac{a+d}{2}\qquad \textbf{(E) }b^2=ad$
$ABCDE$ is a regular pentagon. $AP, AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD, CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then $AO + AQ + AR$ equals
1986 AHSME Problems/Problem 28
$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 1 + \sqrt{5}\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 2 + \sqrt{5}\qquad \textbf{(E)}\ 5$
#Regular_Polygon_Area_Formula
The area of an n-sided regular polygon: The area of an n-sided regular polygon with its side length, s, and apothem, a, is given by:
ABCD is a square with side of unit length. Points E and F are taken respectively on sides AB and AD so that AE = AF and the quadrilateral CDFE has maximum area. In square units this maximum area is:
(A) $\frac{1}{2}$ (B) $\frac{9}{16}$ (C) $\frac{19}{32}$ (D) $\frac{5}{8}$ (E) $\frac{2}{3}$
Convex quadrilateral $ABCD$ has $AB=9$ and $CD=12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC=14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?
2009 AMC 10A Problems/Problem 23
$\mathrm{(A)}\ \frac{9}{2} \qquad \mathrm{(B)}\ \frac{50}{11} \qquad \mathrm{(C)}\ \frac{21}{4} \qquad \mathrm{(D)}\ \frac{17}{3} \qquad \mathrm{(E)}\ 6$
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$?
2009 AMC 10B Problems/Problem 17
$\textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac35\qquad \textbf{(C)}\ \frac23\qquad \textbf{(D)}\ \frac34\qquad \textbf{(E)}\ \frac45$