Given regular pentagon \(ABCDE\), a circle can be drawn that is tangent to \(\overline{DC}\) at \(D\) and to \(\overline{AB}\) at \(A\). The number of degrees in minor arc \(AD\) is $\textbf{(A)}\ 72 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 144$ [asy] size(100); defaultpen(linewidth(0.7)); draw(rotate(18)*polygon(5)); real x=0.6180339887; draw(Circle((-x,0), 1)); int i; for(i=0; i<5; i=i+1) { dot(origin+1*dir(36+72*i)); } label("\(B\)", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("\(A\)", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("\(E\)", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); label("\(D\)", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); label("\(C\)", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));[/asy]
Solution: E
Tags: geometry,isoceles-triangle,trigonometry
Elo: 1485.4358728782763
Source: AMC 12 1995 A Q17
EditTriangle \( ABC\) has a right angle at \( B\), \( AB = 1\), and \( BC = 2\). The bisector of \( \angle BAC\) meets \( \overline{BC}\) at \( D\). What is \( BD\)? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,1), B=(0,0), C=(2,0); pair D=extension(A,bisectorpoint(B,A,C),B,C); pair[] ds={A,B,C,D}; dot(ds); draw(A--B--C--A--D); label("\(1\)",midpoint(A--B),W); label("\(B\)",B,SW); label("\(D\)",D,S); label("\(C\)",C,SE); label("\(A\)",A,NW); draw(rightanglemark(C,B,A,2));[/asy]$$ \textbf{(A)}\ \frac {\sqrt3 - 1}{2} \qquad \textbf{(B)}\ \frac {\sqrt5 - 1}{2} \qquad \textbf{(C)}\ \frac {\sqrt5 + 1}{2} \qquad \textbf{(D)}\ \frac {\sqrt6 + \sqrt2}{2}$$ \( \textbf{(E)}\ 2\sqrt3 - 1\)
Solution: B
Tags: geometry,trigonometry,angle-bisector-theorem
Elo: 1545.8278203282016
Source: AMC 10 2009 B Q20
EditTwo of the altitudes of the scalene triangle \(ABC\) have length \(4\) and \(12\). If the length of the third altitude is also an integer, what is the biggest it can be? $$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $$
Solution: B
Tags: angles,congruent-triangles,geometry,right-triangle,trigonometry,triangle
Elo: 1516.0
Source: AMC 12 1986 A Q29
EditA three-quarter sector of a circle of radius \(4\) inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? [asy] draw(Arc((0,0), 4, 0, 270)); draw((0,-4)--(0,0)--(4,0)); label("\(4\)", (2,0), S); [/asy] $$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$$
Solution: C
Tags: 3d-geometry,volume,area,arc-length,circles,trigonometry,geometry
Elo: 1484.0
Source: AMC 12 2020 B Q9
EditTriangle \( ABC\) has \( AB=13\) and \( AC=15\), and the altitude to \( \overline{BC}\) has length \( 12\). What is the sum of the two possible values of \( BC\)? $$\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19$$
Solution: D
Tags: area-of-triangle,pythagorean-theorem,triangle,geometry,trigonometry,right-triangle
Elo: 1484.0
Source: AMC 12 2009 B Q13
EditQuadrilateral \(ABCD\) satisfies \(\angle ABC = \angle ACD = 90^{\circ}, AC = 20\), and \(CD = 30\). Diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at point \(E\), and \(AE = 5\). What is the area of quadrilateral \(ABCD\)? $$\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$$
Solution: D
Tags: geometry,area,quadrilateral,right-triangle,trigonometry,pythagorean-theorem,angles
Elo: 1516.0
Source: AMC 12 2020 A Q18
EditThe ratio of the radii of two concentric circles is \(1:3\). If \(\overline{AC}\) is a diameter of the larger circle, \(\overline{BC}\) is a chord of the larger circle that is tangent to the smaller circle, and \(AB = 12\), then the radius of the larger circle is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=3*dir(180), B=3*dir(140), C=3*dir(0); dot(O); draw(Arc(origin,1,0,360)); draw(Arc(origin,3,0,360)); draw(A--B--C--A); label("\(A\)", A, dir(O--A)); label("\(B\)", B, dir(O--B)); label("\(C\)", C, dir(O--C)); [/asy] $$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $$
Solution: B
Tags: circles,circle-theorems,trigonometry,geometry
Elo: 1484.0
Source: AMC 12 1992 A Q11
EditAmy painted a dart board over a square clock face using the "hour positions" as boundaries. [See figure.] If \(t\) is the area of one of the eight triangular regions such as that between \(12\) o'clock and \(1\) o'clock, and \(q\) is the area of one of the four corner quadrilaterals such as that between \(1\) o'clock and \(2\) o'clock, then \(\frac{q}{t}=\) [asy] size((80)); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--(.9,0)--(3.1,4)--(.9,4)--(3.1,0)--(2,0)--(2,4)); draw((0,3.1)--(4,.9)--(4,3.1)--(0,.9)--(0,2)--(4,2)); [/asy] $$ \textbf{(A)}\ 2\sqrt{3}-2 \qquad\textbf{(B)}\ \frac{3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(D)}\ \sqrt{3} \qquad\textbf{(E)}\ 2 $$
Solution: A
Tags: area,area-of-triangle,geometry,angles,trigonometry
Elo: 1516.0
Source: AMC 12 1993 A Q17
EditLet \(S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}\), and \((x,y) \neq (0,0) \}\). Let \(T\) be the set of all right triangles whose vertices are in \(S\). For every right triangle \(t=\triangle ABC\) with vertices \(A\), \(B\), and \(C\) in counter-clockwise order and right angle at \(A\), let \(f(t)= \tan (\angle CBA)\). What is \[ \displaystyle \prod_{t \in T} f(t) \text{?} \] [asy] size((120)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); label("\(\circ\)", (0,0)); label("\(S\)", (-.7,2.5)); [/asy] $$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$$
Solution: B
Tags: coordinate-geometry,trigonometry,angles,pythagorean-triples,geometry,coordinates,product,tangent
Elo: 1516.0
Source: AMC 12 2012 B Q25
EditThe angle bisector of the acute angle formed at the origin by the graphs of the lines \(y=x\) and \(y=3x\) has equation \(y=kx\). What is \(k\)? $$\textbf{(A)} \: \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \: \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \: \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \: 2\qquad \textbf{(E)} \: \frac{2+\sqrt{5}}{2}$$
Solution: A
Tags: angle-bisector,angle-bisector-theorem,angles,coordinate-geometry,geometry,trigonometry
Elo: 1484.0
Source: AMC 12 2021 Fall A Q13
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