本讲义涵盖主题 4,几何(Geometry)。标记 (Extended) 的部分只在拓展卷上考查;其他一切对两个层次都适用。在考试中你必须用下面正确的名称给出理由,不只是数字。
几何
IGCSE 数学 · 第 4 主题
4.1
几何术语
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • parallel • perpendicular • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • simple solids. | Includes the following terms: Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. |
| Simple solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere (term ‘hemisphere’ not required) • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • arc • sector • segment. |
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • plane • parallel • perpendicular • perpendicular bisector • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • solids. | Includes the following terms. Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. |
| Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. Solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere • hemisphere • frustum • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • major and minor arc • sector • segment. |
来源:剑桥国际大纲
两条线相遇的一个角是一个顶点(vertex)。两条线平行(parallel)若它们从不相遇,而垂直(perpendicular)若它们以一个直角相遇。角按它们的大小命名:
| 名称 | 大小 |
|---|---|
| 锐角(acute angle) | 小于 $90^{\circ}$ |
| 直角(right angle) | 恰好 $90^{\circ}$ |
| 钝角(obtuse angle) | 在 $90^{\circ}$ 和 $180^{\circ}$ 之间 |
| 优角(reflex angle) | 在 $180^{\circ}$ 和 $360^{\circ}$ 之间 |
我们用三个字母命名一个角,例如角 $ABC$ 是 $B$ 处的角。
探索
Shape and angle lab
Classify angle facts by the diagram feature that creates them.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| vertex | 顶点 | dǐng diǎn |
| parallel | 平行 | píng xíng |
| perpendicular | 垂直 | chuí zhí |
| acute angle | 锐角 | ruì jiǎo |
| right angle | 直角 | zhí jiǎo |
| obtuse angle | 钝角 | dùn jiǎo |
| reflex angle | 优角 | yōu jiǎo |
4.6
角
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of three-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular polygons. | Includes exterior and interior angles, and angle sum. |
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of 3-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular and irregular polygons. | Includes exterior and interior angles, and angle sum. |
来源:剑桥国际大纲
学这些基本事实。每一个在考试中都是一个有效的理由。
- 一个点处的角加起来等于 $360^{\circ}$。
- 一条直线上的角加起来等于 $180^{\circ}$。
- 对顶角(vertically opposite angles)(由两条相交的线构成)相等。
Worked example. 一条直线上的三个角是 $x$、$50^{\circ}$ 和 $70^{\circ}$。求 $x$。
$$x + 50 + 70 = 180 \;\Rightarrow\; x = 60^{\circ}.$$
探索
Parallel line and polygon lab
Pick the angle rule that unlocks each diagram.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| vertically opposite angles | 对顶角 | duì dǐng jiǎo |
4.6
角
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of three-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular polygons. | Includes exterior and interior angles, and angle sum. |
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of 3-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular and irregular polygons. | Includes exterior and interior angles, and angle sum. |
来源:剑桥国际大纲
当一条线穿过两条平行线时:
- 同位角(corresponding angles)(在匹配的位置,一个 "F" 形)相等。
- 内错角(alternate angles)(穿过线的相对两侧,一个 "Z" 形)相等。
- 同旁内角(co-interior angles)(一个 "C" 形)加起来等于 $180^{\circ}$;我们说它们互补(supplementary)。
Worked example. 一条直线穿过两条平行线。一个角是 $110^{\circ}$。同旁内角 $y$ 满足 $110 + y = 180$,所以 $y = 70^{\circ}$。
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| corresponding angles | 同位角 | tóng wèi jiǎo |
| alternate angles | 内错角 | nèi cuò jiǎo |
| co-interior angles | 同旁内角 | tóng páng nèi jiǎo |
| supplementary | 互补 | hù bǔ |
| interior angle | 内角 | nèi jiǎo |
4.1
几何术语
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • parallel • perpendicular • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • simple solids. | Includes the following terms: Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. |
| Simple solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere (term ‘hemisphere’ not required) • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • arc • sector • segment. |
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • plane • parallel • perpendicular • perpendicular bisector • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • solids. | Includes the following terms. Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. |
| Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. Solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere • hemisphere • frustum • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • major and minor arc • sector • segment. |
来源:剑桥国际大纲
一个三角形(triangle)有三条边和加起来等于 $180^{\circ}$ 的角。
| 类型 | 性质 |
|---|---|
| 等边(equilateral) | 所有边相等、所有角 $60^{\circ}$ |
| 等腰(isosceles) | 两条边相等、两个角相等 |
| 不等边(scalene) | 所有边和角都不同 |
| 直角 | 有一个直角 |
Worked example. 一个三角形有角 $x$、$2x$ 和 $90^{\circ}$。求 $x$。
$$x + 2x + 90 = 180 \;\Rightarrow\; 3x = 90 \;\Rightarrow\; x = 30^{\circ}.$$
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| triangle | 三角形 | sān jiǎo xíng |
| equilateral | 等边 | děng biān |
| isosceles | 等腰 | děng yāo |
| scalene | 不等边 | bù děng biān |
4.1
几何术语
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • parallel • perpendicular • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • simple solids. | Includes the following terms: Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. |
| Simple solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere (term ‘hemisphere’ not required) • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • arc • sector • segment. |
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • plane • parallel • perpendicular • perpendicular bisector • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • solids. | Includes the following terms. Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. |
| Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. Solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere • hemisphere • frustum • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • major and minor arc • sector • segment. |
来源:剑桥国际大纲
一个四边形(quadrilateral)有四条边和加起来等于 $360^{\circ}$ 的角。
| 形状 | 性质 |
|---|---|
| 正方形(square) | 四条相等的边、四个直角 |
| 矩形(rectangle) | 对边相等、四个直角 |
| 平行四边形(parallelogram) | 对边平行且相等 |
| 菱形(rhombus) | 四条相等的边、对边平行 |
| 鸢形(kite) | 两对相邻的相等的边 |
| 梯形(trapezium) | 一对平行的边 |
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| quadrilateral | 四边形 | sì biān xíng |
| square | 正方形 | zhèng fāng xíng |
| rectangle | 矩形 | jǔ xíng |
| parallelogram | 平行四边形 | píng xíng sì biān xíng |
| rhombus | 菱形 | líng xíng |
| kite | 鸢形 | yuān xíng |
| trapezium | 梯形 | tī xíng |
4.6
角
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of three-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular polygons. | Includes exterior and interior angles, and angle sum. |
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of 3-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular and irregular polygons. | Includes exterior and interior angles, and angle sum. |
来源:剑桥国际大纲
一个多边形(polygon)是一个有直边的形状。一个正多边形(regular polygon)有所有边和所有角相等。
| 边 | 名称 |
|---|---|
| 5 | 五边形(pentagon) |
| 6 | 六边形(hexagon) |
| 8 | 八边形(octagon) |
| 10 | 十边形(decagon) |
对于一个有 $n$ 条边的多边形:
$$\text{sum of interior angles} = (n - 2) \times 180^{\circ}, \qquad \text{sum of exterior angles} = 360^{\circ}.$$
每个角处的内角(interior angle)和外角(exterior angle)加起来等于 $180^{\circ}$。
Worked example. 求一个正六边形的每个内角。
外角是 $\dfrac{360^{\circ}}{6} = 60^{\circ}$,所以每个内角是 $180^{\circ} - 60^{\circ} = 120^{\circ}$。
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| polygon | 多边形 | duō biān xíng |
| regular polygon | 正多边形 | zhèng duō biān xíng |
| pentagon | 五边形 | wǔ biān xíng |
| hexagon | 六边形 | liù biān xíng |
| octagon | 八边形 | bā biān xíng |
| decagon | 十边形 | shí biān xíng |
| exterior angle | 外角 | wài jiǎo |
4.5
对称
大纲
| Subject content | Notes and examples |
|---|---|
| Recognise line symmetry and order of rotational symmetry in two dimensions. | Includes properties of triangles, quadrilaterals and polygons directly related to their symmetries. |
| Subject content | Notes and examples |
|---|---|
| 1 Recognise line symmetry and order of rotational symmetry in two dimensions. | Includes properties of triangles, quadrilaterals and polygons directly related to their symmetries. |
| 2 Recognise symmetry properties of prisms, cylinders, pyramids and cones. | e.g. identify planes and axes of symmetry. |
来源:剑桥国际大纲
- 一个形状有轴对称(line symmetry)若一条镜像线把它分成两个匹配的一半。
- 一个形状有旋转对称(rotational symmetry)若在你转动它时它落到它自身上。阶数是它在一个整圈里落上多少次。
对于立体(拓展(Extended)),一个把立体分成镜像两半的平坦切片是一个对称面(plane of symmetry),而你能围绕它旋转的一条线是一条对称轴(axis of symmetry)。
探索
Symmetry as a reflection
A shape has line symmetry if reflecting it leaves it unchanged. Reflect the shape and watch what is preserved.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| line symmetry | 轴对称 | zhóu duì chèn |
| rotational symmetry | 旋转对称 | xuán zhuǎn duì chèn |
| plane of symmetry | 对称面 | duì chèn miàn |
| axis of symmetry | 对称轴 | duì chèn zhóu |
4.1
几何术语
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • parallel • perpendicular • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • simple solids. | Includes the following terms: Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. |
| Simple solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere (term ‘hemisphere’ not required) • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • arc • sector • segment. |
| Subject content | Notes and examples |
|---|---|
| 1 Use and interpret the following geometrical terms: • point • vertex • line • plane • parallel • perpendicular • perpendicular bisector • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor. | Candidates are not expected to show that two shapes are congruent. |
| 2 Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • nets • solids. | Includes the following terms. Triangles: • equilateral • isosceles • scalene • right-angled. Quadrilaterals: • square • rectangle • kite • rhombus • parallelogram • trapezium. |
| Polygons: • regular and irregular polygons • pentagon • hexagon • octagon • decagon. Solids: • cube • cuboid • prism • cylinder • pyramid • cone • sphere • hemisphere • frustum • face • surface • edge. | |
| 3 Use and interpret the vocabulary of a circle. | Includes the following terms: • centre • radius (plural radii) • diameter • circumference • semicircle • chord • tangent • major and minor arc • sector • segment. |
来源:剑桥国际大纲
| 术语 | 含义 |
|---|---|
| 圆(circle) | 与一个圆心距离相同的所有点 |
| 圆心(centre) | 中间的点 |
| 半径(radius) | 圆心到边缘(复数 radii) |
| 直径(diameter) | 直接穿过圆心($= 2 \times$ 半径) |
| 圆周(circumference) | 绕一整圈的距离 |
| 弦(chord) | 连接圆上两点的一条直线 |
| 弧(arc) | 圆周的一部分 |
| 扇形(sector) | 两条半径之间的一个"披萨片" |
| 弓形(segment) | 被一条弦切下的区域 |
| 半圆(semicircle) | 半个圆 |
| 切线(tangent) | 在一个点触及圆的一条线 |
探索
Arc and sector
Drag the angle and radius to see the arc (part of the circumference) and the sector (pie slice) it cuts off.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| circle | 圆 | yuán |
| centre | 圆心 | yuán xīn |
| radius | 半径 | bàn jìng |
| diameter | 直径 | zhí jìng |
| circumference | 圆周 | yuán zhōu |
| chord | 弦 | xián |
| arc | 弧 | hú |
| sector | 扇形 | shàn xíng |
| segment | 弓形 | gōng xíng |
| semicircle | 半圆 | bàn yuán |
| tangent | 切线 | qiè xiàn |
4.7 4.8
圆的定理
大纲
| Subject content | Notes and examples |
|---|---|
| Calculate unknown angles and give explanations using the following geometrical properties of circles: • angle in a semicircle = 90° • angle between tangent and radius = 90°. | Candidates will be expected to use the geometrical properties listed in the syllabus when giving reasons for answers. |
| Subject content | Notes and examples |
|---|---|
| Calculate unknown angles and give explanations using the following geometrical properties of circles: • angle in a semicircle = 90° • angle between tangent and radius = 90° • angle at the centre is twice the angle at the circumference • angles in the same segment are equal • opposite angles of a cyclic quadrilateral sum to 180° (supplementary) • alternate segment theorem. | Candidates are expected to use the geometrical properties listed in the syllabus when giving reasons for answers. |
| Subject content | Notes and examples |
|---|---|
| Use the following symmetry properties of circles: • equal chords are equidistant from the centre • the perpendicular bisector of a chord passes through the centre • tangents from an external point are equal in length. | Candidates are expected to use the geometrical properties listed in the syllabus when giving reasons for answers. |
来源:剑桥国际大纲
用这些来求未知的角,总是给出理由。
对于两个层次:
- 半圆里的角是 $90^{\circ}$。
- 一条切线和一条半径之间的角是 $90^{\circ}$。
拓展(定理 I):
- 圆心处的角是圆周处角的两倍(站在同一条弧上)。
- 同一弓形里的角相等。
- 一个圆内接四边形(cyclic quadrilateral)的对角加起来等于 $180^{\circ}$。
- 弦切角定理(alternate segment theorem):一条切线和一条弦之间的角等于另一个弓形里的角。
拓展(定理 II): 相等的弦与圆心距离相同;一条弦的垂直平分线通过圆心;从同一个外部点的两条切线长度相等。
Worked example. A、B、C 在一个圆上。圆周处的角 $ABC$ 是 $40^{\circ}$。求圆心 $O$ 处的角 $AOC$。
圆心处的角是圆周处角的两倍:$2 \times 40^{\circ} = 80^{\circ}$。
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| cyclic quadrilateral | 圆内接四边形 | yuán nèi jiē sì biān xíng |
| alternate segment theorem | 弦切角定理 | xián qiē jiǎo dìng lǐ |
4.4
相似
大纲
| Subject content | Notes and examples |
|---|---|
| Calculate lengths of similar shapes. |
| Subject content | Notes and examples |
|---|---|
| 1 Calculate lengths of similar shapes. | |
| 2 Use the relationships between lengths and areas of similar shapes and lengths, surface areas and volumes of similar solids. | Includes use of scale factor, e.g. $$\frac{\text{Volume of } A}{\text{Volume of } B} = \frac{(\text{Length of } A)^3}{(\text{Length of } B)^3}$$ |
| 3 Solve problems and give simple explanations involving similarity. | Includes showing that two triangles are similar using geometric reasons. |
来源:剑桥国际大纲
两个形状相似(similar)若一个是另一个的一个放大:相同的角,而所有边乘以相同的比例因子(scale factor)$k$。(尺寸和形状完全相同的形状是全等(congruent)。)
对于相似的形状和立体:
$$\frac{\text{area of } A}{\text{area of } B} = k^{2}, \qquad \frac{\text{volume of } A}{\text{volume of } B} = k^{3}.$$
Worked example. 两个相似的立体有比 $2 : 3$ 的长度。较小的有体积 $40\text{ cm}^{3}$。求较大的体积。
体积比是 $2^{3} : 3^{3} = 8 : 27$。所以较大的体积是 $40 \times \dfrac{27}{8} = 135\text{ cm}^{3}$。
探索
Similar shapes — enlargement
Similar shapes are the same shape but a different size. An enlargement scales every length by the same factor; angles stay the same.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| similar | 相似 | xiāng sì |
| scale factor | 比例因子 | bǐ lì yīn zi |
| congruent | 全等 | quán děng |
4.6
角
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of three-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular polygons. | Includes exterior and interior angles, and angle sum. |
| Subject content | Notes and examples |
|---|---|
| 1 Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°. | Knowledge of 3-letter notation for angles is required, e.g. angle $ABC$. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. |
| 2 Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary). | |
| 3 Know and use angle properties of regular and irregular polygons. | Includes exterior and interior angles, and angle sum. |
来源:剑桥国际大纲
一个方位角(bearing)把一个方向作为一个三位数的角给出,从北顺时针(clockwise)测量,从 $000^{\circ}$ 到 $360^{\circ}$。所以正东是 $090^{\circ}$ 而正南是 $180^{\circ}$。
Worked example. $B$ 从 $A$ 的方位角是 $025^{\circ}$。求 $A$ 从 $B$ 的方位角。
返回(反向)方位角相差 $180^{\circ}$:$025^{\circ} + 180^{\circ} = 205^{\circ}$。
探索
Bearings route
Follow how to measure a bearing correctly from north.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| bearing | 方位角 | fāng wèi jiǎo |
| clockwise | 顺时针 | shùn shí zhēn |
4.2 4.3
几何作图
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Measure and draw lines and angles. | A ruler should be used for all straight edges. Constructions of perpendicular bisectors and angle bisectors are not required. |
| 2 Construct a triangle, given the lengths of all sides, using a ruler and pair of compasses only. | e.g. construct a rhombus by drawing two triangles. Construction arcs must be shown. |
| 3 Draw, use and interpret nets. | Examples include: • draw nets of cubes, cuboids, prisms and pyramids • use measurements from nets to calculate volumes and surface areas. |
| Subject content | Notes and examples |
|---|---|
| 1 Draw and interpret scale drawings. | A ruler must be used for all straight edges. |
| 2 Use and interpret three-figure bearings. | Bearings are measured clockwise from north (000° to 360°). e.g. find the bearing of A from B if the bearing of B from A is 025°. |
| Includes an understanding of the terms north, east, south and west. e.g. point D is due east of point C. |
| Subject content | Notes and examples |
|---|---|
| 1 Draw and interpret scale drawings. | A ruler must be used for all straight edges. |
| 2 Use and interpret three-figure bearings. | Bearings are measured clockwise from north ($000^{\circ}$ to $360^{\circ}$). e.g. find the bearing of $A$ from $B$ if the bearing of $B$ from $A$ is $025^{\circ}$. Includes an understanding of the terms north, east, south and west. e.g. point $D$ is due east of point $C$. |
来源:剑桥国际大纲
- 要从三条给定的边作图(construct)一个三角形,用一把尺子画底边,然后用一副圆规(compasses)把每条其他边标记为一条弧。让作图弧显示出来。
- 一个展开图(net)是一个折叠成一个立体的平坦形状。你能用一个展开图算出表面积。
- 一个比例图(scale drawing)以一个固定的比例更小(或更大)地显示一个真实物体,例如 $1\text{ cm}$ 对 $5\text{ m}$。
常见的几何体(solids)及其部分:
| 立体 | 备注 |
|---|---|
| 立方体(cube) / 长方体(cuboid) | 盒子形状 |
| 棱柱(prism) | 沿它整个长度相同的形状 |
| 圆柱(cylinder) | 一个圆形棱柱 |
| 棱锥(pyramid) / 圆锥(cone) | 收到一个点 |
| 球(sphere) / 半球(hemisphere) | 一个球 / 半个球 |
| 平截头体(frustum) | 一个顶部被切掉的圆锥或棱锥 |
一个立体的一个平坦的边是一个面(face),两个面在一条棱(edge)相遇,而整个外部是它的表面(surface)。
探索
Construction and solid lab
Classify geometry tasks by the tool or representation needed.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| construct | 作图 | zuò tú |
| compasses | 圆规 | yuán guī |
| net | 展开图 | zhǎn kāi tú |
| scale drawing | 比例图 | bǐ lì tú |
| solids | 几何体 | jǐ hé tǐ |
| cube | 立方体 | lì fāng tǐ |
| cuboid | 长方体 | cháng fāng tǐ |
| prism | 棱柱 | léng zhù |
| cylinder | 圆柱 | yuán zhù |
| pyramid | 棱锥 | léng zhuī |
| cone | 圆锥 | yuán zhuī |
| sphere | 球 | qiú |
| hemisphere | 半球 | bàn qiú |
| frustum | 平截头体 | píng jié tóu tǐ |
| face | 面 | miàn |
| edge | 棱 | léng |
| surface | 表面 | biǎo miàn |
4.2 4.3
考试技巧
- 一条直线上的角加起来等于 $180°$、一个点周围等于 $360°$,而一个三角形里等于 $180°$。为一道角题的每一步给出一个理由。
- 学圆定理:圆心处的角是圆周处角的两倍;半圆里的角是 $90°$;同一弓形里的角相等;一个圆内接四边形的对角加起来等于 $180°$。
- 任何多边形的外角加起来等于 $360°$,而每个内角 + 它的外角 = $180°$。
- 方位角从北顺时针测量并总是用三位数写(例如 $072°$)。
本主题的互动课程
逐步学习,并即时检测练习。