M203 课程目录

M203 20260328 Polygon

▶ 跳转到 2:42

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

$\textbf{(A) }d=a+b\qquad \textbf{(B) }d^2=a^2+b^2\qquad \textbf{(C) }d^2=a^2+ab+b^2\qquad\\ \textbf{(D) }b=\frac{a+d}{2}\qquad \textbf{(E) }b^2=ad$

▶ 跳转到 10:16

$ABCDE$ 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

$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 1 + \sqrt{5}\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 2 + \sqrt{5}\qquad \textbf{(E)}\ 5$

▶ 跳转到 13:26

#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:

▶ 跳转到 23:15

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) $\frac{2}{3}$

▶ 跳转到 27:40

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

$\mathrm{(A)}\ \frac{9}{2} \qquad \mathrm{(B)}\ \frac{50}{11} \qquad \mathrm{(C)}\ \frac{21}{4} \qquad \mathrm{(D)}\ \frac{17}{3} \qquad \mathrm{(E)}\ 6$

▶ 跳转到 36:25

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

$\textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac35\qquad \textbf{(C)}\ \frac23\qquad \textbf{(D)}\ \frac34\qquad \textbf{(E)}\ \frac45$

▶ 跳转到 46:26

Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$?

2010 AMC 10A Problems/Problem 19

$\textbf{(A)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(B)} \frac{10}{3} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{17}{4} \qquad \textbf{(E)}\ 6$

The area of an equilateral triangle with side length $s$ is $\text{Area} = \frac{\sqrt{3}}{4}s^2$

▶ 跳转到 50:40

Rhombus $ABCD$ has side length $2$ and $\angle B = 120$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?

2011 AMC 10B Problems/Problem 20

$\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad\textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad\textbf{(E)}\ 2$

▶ 跳转到 1:04:48

Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. What is $p+q+r+s$?

2013 AMC 10A Problems/Problem 18

$\textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qquad\textbf{(E)}\ 75$

▶ 跳转到 1:12:14

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?

2013 AMC 10A Problems/Problem 20

$\textbf{(A)}\ 1 - \frac{\sqrt2}{2} + \frac{\pi}{4}\qquad\textbf{(B)}\ \frac{1}{2} + \frac{\pi}{4} \qquad\textbf{(C)}\ 2 - \sqrt2 + \frac{\pi}{4}\qquad\textbf{(D)}\ \frac{\sqrt2}{2} + \frac{\pi}{4} \qquad\textbf{(E)}\ 1 + \frac{\sqrt2}{4} + \frac{\pi}{8}$

▶ 跳转到 1:17:30

In rectangle $ABCD$, $\overline{AB}=20$ and $\overline{BC}=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $\overline{AE}$?

$\textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20$

Note that $\tan 15^\circ=2-\sqrt{3}=\frac{EC}{10} \Rightarrow EC=20-10 \sqrt 3$. (It is important to memorize the sin, cos, and tan values of $15^\circ$ and $75^\circ$.) Therefore, we have $DE=10\sqrt 3$. Since $ADE$ is a $30-60-90$ triangle, $AE=2 \cdot AD=2 \cdot 10=\boxed{\textbf{(E)} \: 20}$

2014 AMC 10A Problems/Problem 22 also refer to others solutions

• 2 Solution 1 (Trigonometry))
• 3 Solution 2 (No Trigonometry))
• 4 Solution 3 Quick Construction (No Trigonometry))
• 5 Solution 4 (No Trigonometry))
• 6 Solution 5
• 7 Solution 6 (Pure Euclidian Geometry))
• 8 Solution 7 (Pure Euclidian Geometry))
• 9 Solution 8 (Trigonometry))
• 10 Solution 9
• 11 Solution 10 Law of Cosines
• 12 Solution 11 (No Trigonometry))
• 13 Solution 12 (Tangent of Sum of Two Angles Formula))
• 14 Solution 13 (constructionusing allowed compass))

▶ 跳转到 1:31:10

A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $\frac{B}{A}$?

2014 AMC 10A Problems/Problem 23

$\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{5}\qquad\textbf{(C)}\ \frac{2}{3}\qquad\textbf{(D)}\ \frac{3}{4}\qquad\textbf{(E)}\ \frac{4}{5}$

▶ 跳转到 1:34:58

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG + JH + CD$?

2015 AMC 10B Problems/Problem 22

$\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10}$

▶ 跳转到 1:42:51

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?

2016 AMC 10A Problems/Problem 24

$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$

• 2 Solution 1
• 3 Solution 2 (Algebra))
• 4 Solution 3 (Extended Law of Sines and Light Trig))
• 5 Solution 4 (Trigonometry Bash))
• 6 Solution 5 (Easier Trigonometry))
• 7 Solution 6 (Ptolemy's Theorem))
• 8 Solution 7 (Trigonometry))
• 9 Solution 8 (Area By Brahmagupta's Formula))
• 10 Solution 9 (Similar Triangles))
• 11 Solution 10 (Complex Numbers))
• 12 Solution 11 (Trignometry + Ptolemy’s))
• 13 Solution 12 (Simple Trigonometry with Geometric Observations))
• 14 Solution 13 (Law of Cosines))
• 15 Solution 14 (Kites))

▶ 跳转到 1:57:47

Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline {AB}$, $\overline{CD}$, and $\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$?

2018 AMC 10B Problems/Problem 24

$\textbf{(A)}\ \frac {3}{8}\sqrt{3} \qquad \textbf{(B)}\ \frac {7}{16}\sqrt{3} \qquad \textbf{(C)}\ \frac {15}{32}\sqrt{3} \qquad \textbf{(D)}\ \frac {1}{2}\sqrt{3} \qquad \textbf{(E)}\ \frac {9}{16}\sqrt{3}$

• 3 Solution 1 (Area Addition))
• 4 Solution 2 (Area Subtraction))
• 5 Solution 3 (Area Subtraction))
• 6 Solution 4 (Area Subtraction))
• 7 Solution 5 (Partition the Hexagon))
• 8 Solution 6 (Partition the Hexagon))
• 9 Solution 7 (Trigonometry))
• 10 Solution 8 (Linear Transformation))
• 11 Solution 9 (Midsegment Theorem))