Geometry
IGCSE Mathematics Topic 4 13:25 English narration · English + 中文 subtitles burned in
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Transcript
Look closely at a honeycomb.
仔细看看蜂巢。
Every cell is a hexagon — six straight sides, all the same.
每一个小房间都是六边形——六条一样长的直边。
Not squares. Not pentagons. Hexagons.
不是正方形，也不是五边形，而是六边形。
Bees never studied geometry, but the shape they build obeys a rule about angles that you are about to learn — and that same rule earns marks on every exam paper.
蜜蜂从没学过几何， 但它们盖出的形状遵守着一条你马上要学的关于角的规则—— 而这条规则在每一份考卷上都能帮你拿分。
This is Topic Four: Geometry.
这是第四个专题：几何。
Angles, triangles, polygons, circles, similar shapes and bearings.
角、三角形、多边形、圆、相似图形和方位角。
It is the topic with the most marks for saying why.
在这个专题里，说清「为什么」能拿到最多的分。
A number on its own is often worth nothing — you must name the reason.
只写一个数字往往一分也拿不到—— 你必须写出理由。
So learn the pictures, and learn the words that go with them.
所以既要记住图，也要记住配套的术语。
Start with the words.
先从术语开始。
An acute angle is less than ninety degrees.
锐角小于九十度。
A right angle is exactly ninety, and we mark it with a small square.
直角正好是九十度，我们用一个小方块来标记它。
An obtuse angle sits between ninety and one hundred and eighty.
钝角在九十度和一百八十度之间。
A reflex angle is bigger than one hundred and eighty.
优角大于一百八十度。
Use the right name every time.
每一次都要用对名称。
Learn four more words first.
先再学四个词。
The corner where two lines meet is a vertex.
两条线相交的那个角点叫做顶点。
Two lines are parallel if they never meet, however far you draw them.
两条线如果怎么延长都永不相交， 它们就平行。
Two lines are perpendicular if they meet at a right angle.
两条线如果相交成直角，它们就垂直。
And we name an angle with three letters: angle A B C is the angle at B, the middle letter.
我们用三个字母给角命名： 角甲乙丙是在乙处的角，中间那个字母就是顶点。
Say the middle letter slowly — that is the vertex you mean.
慢慢读中间那个字母， 那就是你要说的顶点。
Now two facts you will use in almost every question.
接下来是两个几乎每道题都会用到的事实。
Angles on a straight line add up to one hundred and eighty degrees.
同一条直线上的角加起来是一百八十度。
Angles all the way around a point add up to three hundred and sixty.
绕着一个点一圈的角加起来是三百六十度。
Find the angles that share a line or share a point, add up what you know, and subtract from the total.
找出共用一条直线或共用一个点的那些角， 把已知的加起来，再从总数里减掉。
Try a short exam line.
来看一道很短的考题。
Three angles sit on a straight line: x, fifty degrees, and seventy degrees.
三个角在同一条直线上：未知角、五十度、还有七十度。
Find x.
求未知角。
Angles on a straight line add up to one hundred and eighty, so x plus fifty plus seventy equals one hundred and eighty.
同一条直线上的角加起来是一百八十度，所以未知角加五十加七十 等于一百八十。
Fifty and seventy make one hundred and twenty.
五十加七十是一百二十。
So x is sixty degrees.
所以未知角是六十度。
Write the reason: angles on a straight line.
写下理由：同一直线上的角。
Here is the third fact.
这是第三个事实。
Two straight lines crossing make an X shape, and the two angles facing each other are always the same size.
两条直线相交会形成一个 X 形，正对着的两个角总是一样大。
We call them vertically opposite angles, and they stay equal however far you turn the lines.
我们把它们叫做对顶角，无论你把这两条线转多少，它们都保持相等。
Now the habit that saves marks: every time you write an angle, write the reason beside it.
接下来是能保住分数的习惯：每写下一个角，就在旁边写上理由。
When one line crosses two parallel lines, three patterns appear, and the exam loves them.
当一条直线穿过两条平行线时，会出现三种图形，而考试非常喜欢考它们。
Corresponding angles make an F shape, and they are equal.
同位角形成 F 形，它们相等。
Alternate angles make a Z shape, and they are equal too.
内错角形成 Z 形，它们也相等。
Co-interior angles make a C shape, and they add up to one hundred and eighty.
同旁内角形成 C 形，它们加起来是一百八十度。
F equal, Z equal, C adds to one hundred and eighty.
F 相等，Z 相等，C 加起来一百八十度。
Here is the co-interior move in numbers.
用数字看同旁内角。
A straight line crosses two parallel lines.
一条直线穿过两条平行线。
One angle is one hundred and ten degrees.
一个角是一百一十度。
The co-interior angle, y, sits with it in a C shape.
同旁内角乙和它组成一个 C 形。
Co-interior angles are supplementary: they add up to one hundred and eighty.
同旁内角互补：加起来是一百八十度。
So one hundred and ten plus y equals one hundred and eighty.
所以一百一十加乙等于一百八十。
So y is seventy degrees.
所以乙是七十度。
F and Z stay equal; C always adds to one hundred and eighty.
同位角和内错角保持相等；同旁内角永远加到一百八十。
The three angles of a triangle always add up to one hundred and eighty degrees.
三角形的三个角加起来永远是一百八十度。
An equilateral triangle has three equal sides and three angles of sixty.
等边三角形有三条相等的边， 三个角都是六十度。
An isosceles triangle has two equal sides, and the two angles under them are equal — that second half is what students forget.
等腰三角形有两条相等的边，而这两条边下面的两个角也相等—— 后半句正是学生最容易忘的。
A scalene triangle has everything different.
不等边三角形则每条边、每个角都不同。
A quadrilateral has four sides, and its angles add up to three hundred and sixty.
四边形有四条边，它的内角加起来是三百六十度。
One more triangle type: a right-angled triangle has one right angle of ninety degrees.
再补一种三角形：直角三角形有一个九十度的直角。
Now a quick find-x.
现在快速求未知角。
A triangle has angles x, two x, and ninety degrees.
一个三角形的三个角是未知角、二倍未知角、和九十度。
The three angles of a triangle add up to one hundred and eighty, so x plus two x plus ninety equals one hundred and eighty.
三角形三个角加起来 是一百八十度，所以未知角加二倍未知角加九十等于一百八十。
That is three x plus ninety equals one hundred and eighty.
也就是三倍未知角加九十等于一百八十。
Three x is ninety, so x is thirty degrees.
三倍未知角是九十，所以未知角是三十度。
Name the common quadrilaterals by property.
用性质给每一种四边形命名。
A square has four equal sides and four right angles.
正方形四边相等，四个角都是直角。
A rectangle has opposite sides equal and four right angles.
矩形对边相等，四个角都是直角。
A parallelogram has opposite sides parallel and equal.
平行四边形对边平行且相等。
A rhombus has four equal sides and opposite sides parallel.
菱形四边相等，对边平行。
A kite has two pairs of equal sides next to each other.
鸢形有两对相邻的等边。
A trapezium has one pair of parallel sides.
梯形只有一对平行边。
Learn the property, not just the picture — the name alone scores nothing.
要记性质，不要只记样子，只写名称不得分。
Imagine you walk all the way round a polygon, turning at each corner.
想象你绕着一个多边形走一圈，在每个角处转弯。
Back at the start, you have turned one full circle.
回到起点时，你正好转了一整圈。
So the exterior angles of any polygon add up to three hundred and sixty.
所以任何多边形的外角加起来都是三百六十度。
The inside angles are different: their total is the number of sides minus two, times one hundred and eighty.
内角则不同： 内角和等于边数减二，再乘以一百八十度。
For a regular hexagon each exterior angle is sixty, so each interior angle is one hundred and twenty.
对于正六边形，每个外角是六十度， 所以每个内角是一百二十度。
And three hexagons at a point make three hundred and sixty — no gaps.
而三个六边形在一点相接，正好凑成三百六十度—— 没有缝隙。
A polygon is a shape with straight sides.
多边形是由直边围成的图形。
A regular polygon has every side and every angle equal.
正多边形每条边、每个角都相等。
Learn the names: five sides is a pentagon, six a hexagon, eight an octagon, ten a decagon.
记住名称： 五边是五边形，六边是六边形，八边是八边形，十边是十边形。
At each corner the interior angle and the exterior angle add up to one hundred and eighty — they sit on a straight line.
在每个角处，内角和外角加起来是一百八十度，因为它们在同一条直线上。
For a regular hexagon the exterior is sixty, so the interior is one hundred and twenty.
正六边形外角是六十度，所以内角是一百二十度。
Symmetry.
对称。
A shape has line symmetry if a mirror line cuts it into two matching halves — the Taj Mahal has one straight down the middle.
如果一条镜像线把图形分成两个完全一样的半边，这个图形就有轴对称—— 泰姬陵正中就有这样一条线。
A shape has rotational symmetry if it fits onto itself as you turn it, and the order is how many times it fits in one full turn.
如果图形在旋转时能与自身重合，它就有旋转对称， 而阶数就是转一整圈中重合的次数。
For a solid, a slice that gives mirror halves is a plane of symmetry, and a line you can spin it around is an axis of symmetry.
对于立体图形， 把它切成镜像两半的那个平面叫做对称面，能绕着转的那条线叫做对称轴。
Circles have their own vocabulary, and every circle question uses it.
圆有自己的一套术语，每一道圆的题目都会用到。
The radius goes from the centre to the edge.
半径从圆心连到圆上。
The diameter goes right across, through the centre, and is twice the radius.
直径穿过圆心横跨整个圆，等于半径的两倍。
A chord joins two points on the circle.
弦连接圆上的两个点。
A tangent only touches, at one point.
切线只在一个点上碰到圆。
A sector is a slice between two radii, a segment is the piece cut off by a chord, and an arc is part of the edge itself.
扇形是两条半径之间的一块，弓形是被一条弦切下来的部分， 而弧是圆周的一部分。
Two circle theorems are for everybody, for both levels.
有两个圆的定理是所有考生、两个级别都要掌握的。
First: draw a triangle on a diameter, with its third corner on the circle, and the angle at that corner is always ninety degrees.
第一：以直径为一边画一个三角形， 第三个顶点在圆上，那么这个顶点处的角永远是九十度。
That is the angle in a semicircle.
这就是半圆上的圆周角。
Second: a tangent meets a radius at ninety degrees, at the point where it touches.
第二：切线与半径在切点处成九十度。
Two right angles, free — as long as you spot them.
两个直角白送给你——只要你能看出来。
Watch the angle in a semicircle.
看半圆上的圆周角。
The diameter is the long base.
直径是那条长底边。
The third corner slides on the circle.
第三个顶点在圆上滑动。
Whatever place it stops, the angle at that corner is ninety degrees.
无论它停在哪里，那个顶点处的角都是九十度。
So when you spot a triangle on a diameter, mark the right angle straight away and write angle in a semicircle as the reason.
所以一旦看到以直径为一边的 三角形，立刻标出直角，并写上理由：半圆上的圆周角。
Now the Extended theorems, and this is the one examiners love.
现在来看扩展内容的定理，这一条是考官最爱考的。
Take an arc, and look at the angle it makes at the centre and the angle it makes at the edge.
取一段弧， 看它在圆心处形成的角和在圆周上形成的角。
The angle at the centre is always exactly double the angle at the circumference.
圆心角永远正好是圆周角的两倍。
Watch the point move: the shape changes, but double stays double.
看那个点移动：形状在变，但两倍始终是两倍。
Forty at the edge means eighty at the centre.
圆周上是四十度，圆心处就是八十度。
Same rule with numbers.
用数字看同一条规则。
Points A, B and C lie on the circle.
甲、乙、丙三点在圆上。
Angle A B C at the circumference is forty degrees.
圆周角甲乙丙是四十度。
The centre is O, and we want angle A O C.
圆心在圆的正中，要求圆心角。
The angle at the centre is twice the angle at the circumference on the same arc.
同一段弧上，圆心角是圆周角的两倍。
Two times forty is eighty.
二乘四十是八十。
So angle A O C is eighty degrees.
所以圆心角是八十度。
Write the theorem name beside the eighty.
在答案旁边写上定理名称。
Three more Extended theorems.
还有三个扩展内容的定理。
Angles in the same segment are equal: two points on the same side of a chord see it at the same angle.
同弓形内的圆周角相等：弦同一侧的两个点， 看这条弦的角度是一样的。
A cyclic quadrilateral has all four corners on the circle, and its opposite angles add up to one hundred and eighty.
圆内接四边形的四个顶点都在圆上， 它的对角加起来是一百八十度。
And the alternate segment theorem: the angle between a tangent and a chord equals the angle in the segment on the other side.
还有弦切角定理： 切线与弦的夹角，等于另一侧弓形内的圆周角。
Three more Extended facts about chords and tangents.
关于弦和切线，还有三条扩展事实。
Equal chords sit the same distance from the centre.
相等的弦到圆心的距离相等。
The perpendicular bisector of a chord always passes through the centre — so drop a right-angle line through the middle of a chord and you have found the centre.
弦的垂直平分线一定经过圆心，所以从弦的中点作垂线，就能找到圆心。
And two tangents from the same outside point are equal in length.
从圆外同一点引出的两条切线长度相等。
Spot either picture and you have a free equal pair.
认出这两种图，你就白得一对等量。
Two shapes are similar if one is just an enlargement of the other: the same angles, and every length multiplied by the same scale factor.
如果一个图形只是另一个图形的放大，两个图形就是相似的：角相同， 每条边长都乘以同一个比例因子。
Now be careful.
现在要小心。
Lengths scale by the factor. Areas scale by the factor squared. Volumes scale by the factor cubed.
长度按这个因子变化， 面积按因子的平方变化，体积按因子的立方变化。
Lengths in the ratio two to three means volumes in the ratio eight to twenty-seven.
长度之比是二比三，体积之比就是八比二十七。
Nesting dolls show similarity: same shape, different size.
套娃体现相似：形状相同，大小不同。
Shapes that are the same size and the same shape are congruent — a stronger match than similar.
大小和形状都完全一样的图形是全等—— 比相似更强。
Now a volume question.
现在看一道体积题。
Two similar solids have lengths in the ratio two to three.
两个相似立体的长度比是二比三。
The smaller volume is forty cubic centimetres.
较小的体积是四十立方厘米。
The volume ratio is eight to twenty-seven.
体积比是八比二十七。
So the larger volume is forty times twenty-seven over eight, which is one hundred and thirty-five cubic centimetres.
所以较大的体积是四十 乘以二十七再除以八，等于一百三十五立方厘米。
A bearing is a direction written with three figures.
方位角是用三位数写出的方向。
Always start at north, and always turn clockwise.
永远从北开始，永远按顺时针方向转。
East is zero nine zero, south is one eight zero, west is two seven zero.
东是零九零，南是一八零，西是二七零。
Twenty-five degrees is written zero two five — keep that leading zero.
二十五度要写成零二五—— 前面的零不能省。
And the bearing back the other way differs by one hundred and eighty, so zero two five becomes two zero five.
反过来的回方位角相差一百八十度， 所以零二五就变成二零五。
Unfold a solid and you get its net: the flat shape that folds back up.
把立体图形展开就得到它的展开图：能折回去的那个平面图形。
A cube opens into six equal squares.
正方体展开成六个相等的正方形。
A cylinder opens into two circles and one rectangle, and that rectangle is as long as the circumference of the circle.
圆柱展开成两个圆和一个长方形， 而这个长方形的长就等于圆的周长。
Add up the pieces and you have the surface area.
把各部分的面积加起来，就是表面积。
Two practical skills.
两项实用技能。
First, construct a triangle from three sides: draw the base with a ruler, then open a pair of compasses to each other side and swing an arc from each end of the base.
第一，用三边作三角形：先用直尺画出底边，再用圆规分别 按另两边的长度，从底边两端画弧。
Where the arcs cross is the third corner.
两弧的交点就是第三个顶点。
Leave the construction arcs showing — the examiner wants to see them.
留下作图弧线，考官要看到它们。
Second: a scale drawing shows a real object smaller or larger by a fixed scale, such as one centimetre to five metres.
第二：比例图按固定比例把实物画得更小 或更大，比如一厘米代表五米。
Name the common solids.
记住常见的立体。
A cube and a cuboid are box shapes.
立方体和长方体都是盒子形。
A prism keeps the same shape all along its length, and a cylinder is a circular prism.
棱柱沿长度方向截面形状不变， 圆柱就是圆截面的棱柱。
A pyramid and a cone come to a point.
棱锥和圆锥收成一点。
A sphere is a ball; a hemisphere is half a ball.
球是圆球；半球是半个球。
A frustum is a cone or pyramid with the top cut off.
平截头体是切去顶部的圆锥或棱锥。
A flat side is a face, two faces meet at an edge, and the whole outside is the surface.
一个平面侧面叫面，两个面相交成棱， 整个外侧叫表面。
Read the cylinder net carefully.
仔细看圆柱的展开图。
Two circles each have area pi r squared.
两个圆每个面积都是派乘半径的平方。
The rectangle in the middle has width two pi r — that is the circumference — and height h.
中间的长方形 宽是两派半径，也就是圆周长，高是柱高。
So the total surface area is two pi r squared plus two pi r h.
所以总表面积是两派半径平方 加上两派半径乘柱高。
Always add every face.
一定要把每个面都加进去。
Miss a circle and you lose the marks for the surface.
漏掉一个圆，表面积的分 就没了。
The same method works for any solid once you have its net.
只要有展开图，任何立体都能用同样的方法。
One exam question to finish.
最后来看一道考题。
The two long lines are parallel.
两条长直线是平行的。
One angle is sixty-two degrees, another is fifty-five, and we want x.
一个角是六十二度，另一个是五十五度， 要求 x。
Pause here and try it.
先暂停，自己试一试。
Ready?
好了吗？
Always start with the parallel lines.
永远先从平行线入手。
The angle at the bottom left is also sixty-two, because alternate angles are equal.
左下角那个角也是六十二度，因为内错角相等。
Those three angles at the bottom sit on a straight line, so they add up to one hundred and eighty.
底下这三个角在同一条直线上，所以加起来是一百八十度。
So x is sixty-three degrees — and every step has a reason.
所以 x 等于六十三度——而且每一步都有理由。
Three habits before you go.
结束之前，三个习惯。
First, name the reason for every angle you write — alternate, co-interior, vertically opposite, angles in a triangle.
第一，为你写下的每一个角写出理由—— 内错角、同旁内角、对顶角、三角形内角和。
A bare number often scores nothing.
只写一个数字往往一分也拿不到。
Second, the angle at the centre is double the angle at the circumference, not the other way round.
第二，圆心角是圆周角的两倍，不要反过来。
Third, bearings are three figures, measured clockwise from north.
第三，方位角要写成三位数，从北开始按顺时针量。
Get those, and Topic Four is yours.
做到这些，第四个专题就是你的了。