In the figure, parallel lines \(a\) and \(b\) are intersected by lines \(c\), \(d\), and \(e\). If \(z=49\), \(y=136\), and \(v<z\), which statement about \(x\) and \(w\) must be true?
A total of 3 squares each have side length \(r\). A total of 6 equilateral triangles each have side length \(t\). None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is 60. Which equation represents this situation?
The lengths of the two sides of a triangle are 12 inches and 7 inches, respectively. If the length of the third side of the triangle is \(x\) inches such that \(x\) is an integer, how many possible values can \(x\) have?
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 7 and 10, which inequality represents the possible lengths, \(x\), of the third side of the triangle?
A) \(x<17\)
B) \(x>17\)
C) \(3<x<17\)
D) \(x\le17\)
答案与提示
答案：C \(3<x<17\)
提示：上界 \(7+10\)，下界 \(10-7\)——A 只给了上界，忘了"差"那一头。
练 2.2HardCollege Board 题库 · 正确率 71%
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12, which inequality represents the possible lengths, \(x\), of the third side of the triangle?
In triangles LMN and RST, angles L and R each have measure 60°, LN = 10, and RT = 30. Which additional piece of information is sufficient to prove that triangle LMN is similar to triangle RST?
A) MN = 7 and ST = 7
B) MN = 7 and ST = 21
C) The measures of angles M and S are 70° and 60°, respectively.
D) The measures of angles M and T are 70° and 50°, respectively.
In triangle RST, angle T is a right angle, point L lies on \(RS\), point K lies on \(ST\), and \(LK\) is parallel to \(RT\). If the length of \(RT\) is 72 units, the length of \(LK\) is 24 units, and the area of triangle RST is 792 square units, what is the length of \(KT\), in units?
In the figure, \(LQ\) intersects \(MP\) at point R, and \(LM\) is parallel to \(PQ\). The lengths of \(MR\) and \(RP\) are 9 and 17 units, respectively. The area of \(\triangle LMR\) is 45 square units. What is the area of \(\triangle PQR\), in square units?
易错点：C（\(45\times\frac{17}{9}=85\)）是"面积按一次比例缩放"的经典错误——长度 ×k，面积必须 ×k²。X 型（沙漏型）和 A 型（嵌套型，例 3B）是平行线造相似的两种脸，判定逻辑相同。
练习组
独立完成 · 4 题
含例 3A / 3B 的原题变体
练 3.1（例 3A 变体 · 更难）Hard2024年6月C卷 · 正确率 60%
For triangle PQR and triangle STU, angles P and S each measure 30°, QR = 16 and TU = 12. Which additional piece(s) of information is(are) sufficient to prove that triangle PQR is similar to triangle STU?
I. The length of \(PQ\) is \(\dfrac43\) the length of \(ST\). II. The length of \(PR\) is \(\dfrac43\) the length of \(SU\).
In triangle RST, angle T is a right angle, point L lies on \(RS\), point K lies on \(ST\), and \(LK\) is parallel to \(RT\). If the length of \(RT\) is 72 units, the length of \(LK\) is 24 units, and the area of triangle RST is 468 square units, what is the length of \(KT\), in units?
In triangle RST, angle T is a right angle, point L lies on \(RS\), point K lies on \(ST\), and \(LK\) is parallel to \(RT\). If the length of \(RT\) is 63 units, the length of \(LK\) is 21 units, and the area of triangle RST is 252 square units, what is the length of \(KT\), in units?
In triangle ABC, the measure of angle B is 90° and \(BD\) is an altitude of the triangle. The length of \(AB\) is 15 and the length of \(AC\) is 23 greater than the length of \(AB\). What is the value of \(\dfrac{BC}{BD}\)?
The perimeter of an equilateral triangle is 750 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is \( w\sqrt{3} \) centimeters. What is the value of \(w\)?
In right triangle RST, the sum of the measures of angle R and angle S is 90 degrees. The value of \(\sin(R)\) is \( \dfrac{3\sqrt{17}}{13} \). What is the value of \(\cos(S)\)?
A) \(\dfrac{4\sqrt{17}}{51}\)
B) \(\dfrac{3\sqrt{17}}{13}\)
C) \(\dfrac{13\sqrt{17}}{51}\)
D) \(\dfrac{3\sqrt{17}}{4}\)
查看解答
答案：B（原样照抄）
互余 ⇒ \(\cos S=\sin R\)。D 大于 1 可当合理性检查排除。
题型 5B
互余角的 tan：互为倒数
例 5B-1HardCollege Board 题库 · 正确率 85%
In a right triangle, the tangent of one of the two acute angles is \( \dfrac{\sqrt{3}}{3} \). What is the tangent of the other acute angle?
A) \(-\dfrac{\sqrt3}{3}\)
B) \(-\dfrac{3}{\sqrt3}\)
C) \(\dfrac{\sqrt3}{3}\)
D) \(\dfrac{3}{\sqrt3}\)
查看解答
答案：D \(\dfrac{3}{\sqrt3}\)（即 \(\sqrt3\)）
对边邻边互换 ⇒ tan 互为倒数（图 5-1）。锐角 tan 恒正，A、B 直接排除。
题型 5C
由 tan 求边长
例 5C-1Hard2025年3月美国C卷 · 正确率 87%
In right triangle ABC, angles A and B are acute, side AC has a length of 23.3, and \( \tan B=\dfrac{1}{4} \). What is the length of side BC, rounded to the nearest tenth?
A) \(4.8\)
B) \(5.8\)
C) \(93.2\)
D) \(542.9\)
查看解答
答案：C \(93.2\)
直角在 C。对 B：对边 \(AC\)、邻边 \(BC\)。
\(\dfrac{23.3}{BC}=\dfrac14\Rightarrow BC=93.2\)。
易错点：B（÷4）把 AC 当邻边——先画图再认对邻。
题型 5D
相似三角形的三角比转移
例 5D-1HardCollege Board 题库 · 正确率 85%
\( RS=440 \) \( ST=384 \) \( TR=584 \) The side lengths of right triangle RST are given. Triangle RST is similar to triangle UVW, where S corresponds to V and T corresponds to W. What is the value of \(\tan W\)?
In triangle JKL, the measure of angle J is \(90b°\), the measure of angle K is \(66a°\), and the measure of angle L is \(24a°\), where \(a\) and \(b\) are constants. Which of the following must be true?
A) \(\cos L>\sin K\)
B) \(\cos L=\sin K\)
C) \(\cos L<\sin K\)
D) There is not enough information to compare the values of \(\cos L\) and \(\sin K\).
In right triangle RST, the sum of the measures of angle R and angle S is 90 degrees. The value of \(\sin(R)\) is \( \dfrac{4\sqrt{2}}{9} \). What is the value of \(\cos(S)\)?
A) \(\dfrac{4\sqrt2}{9}\)
B) \(\dfrac{4\sqrt2}{7}\)
C) \(\dfrac{7\sqrt2}{8}\)
D) \(\dfrac{9\sqrt2}{8}\)
答案与提示
答案：A \(\dfrac{4\sqrt2}{9}\)
提示：互余 ⇒ \(\cos S=\sin R\)，原样照抄。
练 5.2（例 5A 变体）Hard2024年12月B卷 · 正确率 95%
In right triangle RST, the sum of the measures of angle R and angle S is 90 degrees. The value of \(\sin(R)\) is \( \dfrac{2\sqrt{10}}{7} \). What is the value of \(\cos(S)\)?
A) \(\dfrac{3\sqrt{10}}{20}\)
B) \(\dfrac{2\sqrt{10}}{7}\)
C) \(\dfrac{7\sqrt{10}}{20}\)
D) \(\dfrac{2\sqrt{10}}{3}\)
答案与提示
答案：B \(\dfrac{2\sqrt{10}}{7}\)
提示：同一模板第三次出现——认出"两锐角和 90°"就是秒答题。
练 5.3（例 4A 变体）Hard2024年10月A卷 · 正确率 70%
The perimeter of an isosceles right triangle is \( 28+28\sqrt{2} \) inches. What is the length, in inches, of the hypotenuse of this triangle?
"正角（positive angle）"：从 x 轴正方向逆时针转为正、顺时针为负；加减 \(360°\)（或 \(2\pi\)）不改变终边位置。
图 6-1 ｜ 弧度 = 弧长与半径之比；换算只记 180°=π 一条
交互演示
单位圆与 sin / cos / tan 图像联动
移动鼠标 · 固定点 · 输入角度或弧度
在单位圆或函数图像上移动鼠标观察对应点；单击可固定，也可直接输入度数或弧度。
练习组
独立完成 · 3 题
换算 · 共终边角 · 周期性
练 6.1Medium机考OG1 · 正确率 81%
The measure of angle R is \( \dfrac{2\pi}{3} \) radians. The measure of angle T is \( \dfrac{5\pi}{12} \) radians greater than the measure of angle R. What is the measure of angle T, in degrees?
For angle A, the value of \( \tan A=\dfrac{\sqrt3}{3} \). Which of the following could be the measure of angle A, in radians?
I. \(\dfrac{25\pi}{6}\) II. \(\dfrac{53\pi}{6}\) III. \(\dfrac{77\pi}{6}\)
Point F lies on a unit circle in the xy-plane and has coordinates (1, 0). Point G is the center of the circle and has coordinates (0, 0). Point H also lies on the circle and has coordinates \((-1, y)\), where \(y\) is a constant. Which of the following could be the positive measure of angle FGH, in radians?
In the xy-plane, a unit circle with center at the origin O contains point A with coordinates \((1,\,0)\) and point B with coordinates \( \left(\dfrac{5}{\sqrt{34}},\ -\dfrac{3}{\sqrt{34}}\right) \). If the measure of angle AOB is \(p\) radians, what is the value of \( \dfrac{\cos p}{\sin p} \)?
Point N lies on a unit circle in the xy-plane and has coordinates (1, 0). Point O is the center of the circle and has coordinates (0, 0). Point M also lies on the unit circle, and the measure of angle NOM is \( \dfrac{272\pi}{3} \) radians. If the coordinates of point M are \((a,\ b)\), where \(a\) and \(b\) are constants, what is the value of \(a\)?
\( x^2+y^2-6x-10y-4n=0 \) The given equation represents circle A in the xy-plane, where \(n\) is a constant. Point \((5,7)\) lies on circle B, which has the same center but twice the diameter as circle A. What is the value of \(n\)?
A circle has diameters \(AC\) and \(BD\). The circumference of the circle is \(168\pi\), and the length of arc DA is 3 times the length of arc AB. What is the length of arc BC?
A) \(3\pi\)
B) \(21\pi\)
C) \(28\pi\)
D) \(63\pi\)
查看解答
答案：D \(63\pi\)
BD 是直径 ⇒ 弧 DA + 弧 AB = 半圆 \(=84\pi\)；设弧 AB \(=s\)：\(4s=84\pi\)，\(s=21\pi\)。
A circle has center G, and points M and N lie on the circle. Line segments MH and NH are tangent to the circle at points M and N, respectively. If the radius of the circle is 168 millimeters and the perimeter of quadrilateral GMHN is 3,856 millimeters, what is the distance, in millimeters, between points G and H?
A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. The diagonal of the rectangle is twice the length of the shortest side of the rectangle. The area of the rectangle is \(1{,}089\sqrt{3}\) square units. What is the length, in units, of the diameter of the circle?
A circle is inscribed in a square such that the circumference of the circle touches the midpoint of each side of the square. The length of the diagonal of the square is 228 units. What is the area, in square units, of the circle?
Points A, B, and C lie on a circle with a radius of 15 inches. If the length of arc ABC is \(5\pi\) inches, what is the length, in inches, of chord \(AC\)?
\( x^2+y^2=36 \) \( y=mx+\dfrac{b}{4} \) In the given system of equations, \(m\) and \(b\) are negative constants. In the xy-plane, the graphs of the equations in the given system intersect at the point \((-5,\ y)\), where \(y<0\). Which expression represents the value of \(b\)?
In the figure shown, points A, B, C, and E lie on the circle, and \(AB<BC\). Segment \(AC\) is perpendicular to segment \(BE\) at point D, and \(BD=\sqrt{390}\). The diameter of the circle is 197. If \( \dfrac{CD}{AD}=r \), what is the value of \(r\)?
A right circular cylinder has a volume of \(6{,}591\pi\) cubic inches. The height of the cylinder is 3 times its radius. What is the radius, in inches, of the cylinder?
A right circular cone has a volume of \(34{,}496\pi\) cubic centimeters and the area of its base is \(3{,}136\pi\) square centimeters. What is the slant height, in centimeters, of this cone?
A) \(11\)
B) \(33\)
C) \(56\)
D) \(65\)
查看解答
答案：D \(65\)
底面积 ⇒ \(r=56\)；体积 \(=\frac13(底面积)h\) ⇒ \(h=33\)。
斜高（图 9-1 左）：\(l=\sqrt{56^2+33^2}=65\)。
易错点：B 是高、C 是半径——问的是斜高，三步缺一不可。
题型 9C
棱锥：表面积 → 斜高
例 9C-1Hard2024年5月北美A卷 · 正确率 67% · 填空题
A right square pyramid has a surface area of 75,264 square inches, which includes a base area of 36,864 square inches. What is the slant height, in inches, of this pyramid?
The density of a certain type of wood is 353 kilograms per cubic meter. A sample of this type of wood is in the shape of a cube and has a mass of 345 kilograms. To the nearest hundredth of a meter, what is the length of one edge of this sample?
A hemisphere is half of a sphere. If a hemisphere has a radius of 83 inches, which of the following is closest to the volume, in cubic inches, of this hemisphere?
A right circular cone has a volume of \(4{,}800\pi\) cubic centimeters and the area of its base is \(1{,}600\pi\) square centimeters. What is the slant height, in centimeters, of this cone?
A right square pyramid has a height of 11 centimeters (cm). A second right square pyramid has a height of 22 centimeters. The area of the bases of each of the two pyramids is 100 cm². Which of the following is closest to the difference in the surface area of the second pyramid and the surface area of the first pyramid, in cm²?
The figure shown is a right rectangular pyramid, where \(l=18\) units, \(w=9\) units, and \(h=12\) units. What is the surface area, in square units, of the pyramid?
The figure shown is a right rectangular pyramid, where \(l=16\) units, \(w=8\) units, and \(h=18\) units. What is the surface area, in square units, of the pyramid?
The height of a right circular cylinder is 59 inches, and the circumference of its base is 590 inches. Which expression represents the total surface area, in square inches, of the cylinder?
A) \((59)(590)\)
B) \(\pi\left(\dfrac{295}{\pi}\right)^{2}(59)\)
C) \((59)(590)+\pi\left(\dfrac{295}{\pi}\right)^{2}\)
The height of a right circular cylinder is 47 inches, and the circumference of its base is 376 inches. Which expression represents the total surface area, in square inches, of the cylinder?
A) \(47\cdot376+\pi\left(\dfrac{188}{\pi}\right)^{2}\)
B) \(\pi\left(\dfrac{188}{\pi}\right)^{2}(47)\)
C) \(47\cdot376+2\pi\left(\dfrac{188}{\pi}\right)^{2}\)
In the figure, a hollow steel pipe is in the shape of a right circular cylinder. The steel pipe has an outside diameter of 72 inches, a wall thickness of \(\dfrac58\) inches, and a height of 138 inches. Which of the following is closest to the volume, in cubic inches, of the wall of this steel pipe?
In the figure, a hollow steel pipe is in the shape of a right circular cylinder. The steel pipe has an outside diameter of 76 inches, a wall thickness of \(\dfrac58\) inches, and a height of 135 inches. Which of the following is closest to the volume, in cubic inches, of the wall of this steel pipe?
Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 59 square centimeters (cm²), and the surface area of right rectangular prism Y is 1,475 cm². The volume of right rectangular prism Y is 1,500 cubic centimeters (cm³). What is the sum of the volumes, in cm³, of right rectangular prism X and right rectangular prism Y?
Each side of square A has a length of 13 inches. Each side of square A is multiplied by a scale factor of 3 to create square B. What is the length, in inches, of each side of square B?
A) \(39\)
B) \(16\)
C) \(13\)
D) \(10\)
答案与提示
答案：A \(39\)
提示：scale factor 是"乘"不是"加"（B 是 13+3 的陷阱）。追问自己：B 的面积是 A 的几倍？——\(3^2=9\) 倍。
练 10.2（例 10A 变体）Hard2025年3月美国C卷 · 正确率 72% · 填空题
Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 56 square centimeters (cm²), and the surface area of right rectangular prism Y is 896 cm². The volume of right rectangular prism Y is 1,536 cubic centimeters (cm³). What is the sum of the volumes, in cm³, of right rectangular prism X and right rectangular prism Y?
The table shows the volume of two similar solids, right circular cylinder A and right circular cylinder B. The radius of right circular cylinder A is 3 units. The surface area of cylinder A is \(k\pi\) square units, and the surface area of cylinder B is \(n\pi\) square units, where \(k\) and \(n\) are constants. What is the value of \(n-k\)? (The surface area of a right circular cylinder with radius \(r\) and height \(h\) is \(2\pi r^2+2\pi rh\).)
