Coordinate geometry
IGCSE Mathematics Topic 3 7:35 English narration · English + 中文 subtitles burned in
Chapters
Transcript
Look down on a city like this and every street corner has an address: so many blocks across, so many blocks up.
从空中看这样一座城市，每个街角都有自己的地址：横向几个街区，纵向几个街区。
Two numbers, and you know exactly where you are.
两个数字，就能确定你在哪里。
Mathematics does the same thing on a grid of its own.
数学也在自己的网格上做同样的事。
Give me two numbers and I can find your point, measure how far apart two points are, and write down the line that joins them.
给我两个数字，我就能找到你的点，量出两点之间有多远，还能写出连接它们的那条直线。
This is Topic Three: coordinate geometry.
这是第三个专题：坐标几何。
Everything in it is built from points on a grid and the straight lines through them.
这里的一切都建立在网格上的点， 以及穿过它们的直线之上。
Parts marked Extended are only for the Extended papers.
标注 Extended 的部分只在扩展卷中考查。
Let's begin.
让我们开始吧。
Here is the plan.
先看计划。
First coordinates and the axes.
首先是坐标与坐标轴。
Then the straight line, and what its two numbers mean.
然后是直线，以及它那两个数字的含义。
Then gradient, then length and midpoint, and last of all parallel and perpendicular lines.
接着是斜率，然后是长度与中点，最后是平行线与垂直线。
Five ideas, and each one leans on the one before it.
五个概念，每一个都依赖前一个。
Start with the grid itself.
先从网格本身说起。
The two number lines are the axes: the horizontal one is the x-axis, the vertical one is the y-axis.
这两条数轴叫做坐标轴：水平的是 x 轴，竖直的是 y 轴。
They cross at the origin.
它们相交于原点。
Every point is written as two numbers in brackets, across first and then up.
每个点都写成括号里的两个数，先横向再纵向。
So this point means three to the right, then two down.
所以这个点表示向右三格，再向下两格。
The axes cut the grid into four quadrants, numbered from the top right, going anticlockwise.
坐标轴把网格分成四个象限，从右上角开始，按逆时针方向编号。
Now the straight line.
现在看直线。
Almost every line you meet is written as y equals m x plus c.
你遇到的几乎每条直线都写成 y 等于 m x 加 c。
Only two numbers control it.
只有两个数控制它。
The gradient, m, tilts the line: make it bigger and the line gets steeper.
斜率 m 使直线倾斜：它越大，直线越陡。
The intercept, c, slides the line up and down, and it is the height where the line crosses the y-axis.
截距 c 使直线上下平移，它就是直线与 y 轴相交处的高度。
Watch what each one does on its own.
看看它们各自的作用。
A line does not always arrive in that shape.
直线并不总是以那种形式出现。
Here is five x plus four y equals eight.
这里是五 x 加四 y 等于八。
Rearrange it, and both numbers are yours.
把它整理一下，两个数就都到手了。
First move the x term across: four y equals minus five x plus eight.
先把含 x 的项移到另一边：四 y 等于负五 x 加八。
Then divide every single term by four.
然后把每一项都除以四。
Now read it off: the gradient is minus five over four, and the intercept is two.
现在直接读出来：斜率是负四分之五，截距是二。
Gradient is just steepness, the same steepness you feel on a mountain road.
斜率就是陡峭程度，就像你在盘山公路上感受到的那种陡。
In maths we measure it as rise over run: how far the line goes up, divided by how far it goes across.
在数学里我们用上升量除以水平位移来衡量它：直线向上走了多少，除以向右走了多少。
A big number means a steep climb. A small number means a gentle one.
数值大表示爬得陡，数值小表示比较平缓。
And if the line goes down as you move to the right, the gradient is negative.
如果向右走时直线是向下的，斜率就是负的。
Most questions give you two points instead of a picture.
大多数题目给你两个点，而不是一张图。
Take one and two, and four and eleven.
取一和二，以及四和十一。
The rule is change in y over change in x.
法则是 y 的变化量除以 x 的变化量。
On the top, subtract the y values: eleven take away two is nine.
上面，把两个 y 相减：十一减二等于九。
On the bottom, subtract the x values in the same order: four take away one is three.
下面，按同样的顺序把两个 x 相减：四减一等于三。
So the gradient is nine over three, which is three.
所以斜率是三分之九，也就是三。
Keep the same point first on the top and on the bottom, or the sign comes out wrong.
上下都要从同一个点开始减，否则符号就会算错。
Drawing a line is quick if you use those two numbers.
如果用好这两个数，画直线很快。
Take y equals three x minus one.
取 y 等于三 x 减一。
Mark the intercept first, minus one on the y-axis.
先标出截距：y 轴上的负一。
Then use the gradient: from that point step one to the right and three up, and mark a new point.
然后用斜率：从那个点向右一格、向上三格，标出一个新点。
Do it once more, then join them with a ruler.
再做一次，然后用直尺连起来。
If you want a check, build a small table of values and see the same points appear.
想检验的话，列一个小的数值表，看看是不是同样的点。
Now the reverse job.
现在反过来做。
A line has gradient three and it passes through the point one, two.
一条直线斜率为三，经过点一、二，求它的方程。
Find its equation. Start by writing what you know: y equals three x plus c.
先写下已知的部分：y 等于三 x 加 c。
Then substitute the point, because that point sits on the line: two equals three times one, plus c.
然后把这个点代进去，因为这个点在直线上： 二等于三乘以一，再加 c。
Solve it, and c is minus one.
解出来，c 等于负一。
So the line is y equals three x minus one, the very line we just drew.
所以这条直线是 y 等于三 x 减一，正是我们刚刚画的那一条。
Two Extended tools now, and both come out of one picture.
现在是两个扩展内容的工具，它们都来自同一张图。
Join one, two to four, six.
把一、二和四、六连起来。
The gap across is three and the gap up is four.
横向的间隔是三，纵向的间隔是四。
Those two gaps make a right-angled triangle, so the length comes from Pythagoras' theorem: three squared plus four squared is twenty-five, and the square root of that is five.
这两个间隔构成一个直角三角形， 所以长度用勾股定理求：三的平方加四的平方等于二十五，它的平方根是五。
The midpoint is easier — average the two x values and average the two y values, giving two point five and four.
中点更简单——把两个 x 取平均，把两个 y 取平均，得到二点五和四。
Parallel lines never meet, and that is only possible if they have exactly the same gradient.
平行线永不相交，而这只有在它们斜率完全相同时才可能。
Only the intercept changes.
变的只是截距。
So here is the classic question: find the line parallel to y equals four x minus one that passes through one, minus three.
于是有了这道经典题：求与 y 等于四 x 减一平行、并且经过一、负三的直线。
The gradient stays four.
斜率仍然是四。
Put the point in: minus three equals four times one, plus c, so c is minus seven.
把点代进去：负三等于四乘以一，再加 c，所以 c 等于负七。
The answer is y equals four x minus seven.
答案是 y 等于四 x 减七。
Perpendicular lines are different: they cross at a right angle.
垂直线不一样：它们成直角相交。
On the left, two parallel lines run side by side, sharing one gradient.
左边，两条平行线并排延伸，共用一个斜率。
On the right, the two gradients are two and minus one half.
右边，两个斜率是二和负二分之一。
Multiply those together and you get minus one.
把它们相乘，得到负一。
That is always true for perpendicular lines, and it is the rule the exam tests.
对垂直线来说这永远成立，也正是考试要考的法则。
So how do you find the second gradient?
那么怎样求第二个斜率呢？
The two gradients multiply to minus one, so the second one is minus one over the first.
两个斜率相乘等于负一， 所以第二个就是负的第一个的倒数。
In words: flip the fraction, then change the sign.
用一句话说：把分数翻过来，再变号。
A gradient of two becomes minus one half.
斜率二变成负二分之一。
Three over two becomes minus two thirds.
二分之三变成负三分之二。
Now the exam's own example: two y equals three x plus one.
现在看考试自己给的例子：二 y 等于三 x 加一。
Rearrange it first, so the gradient is three over two, and the perpendicular gradient is minus two thirds.
先整理，斜率是二分之三，所以垂直方向的斜率是负三分之二。
One full question to finish.
最后来做一整道题。
Find the perpendicular bisector of the segment joining minus three, eight and nine, minus two.
求连接负三、八和九、负二这条线段的垂直平分线。
Pause here and try it.
先暂停，自己试一试。
Step one, the midpoint: average the x values, average the y values, and you get three, three.
第一步，中点：把两个 x 取平均，把两个 y 取平均，得到三、三。
Step two, the gradient of the segment is minus ten over twelve, which is minus five over six.
第二步，线段的斜率是负十二分之十，也就是负六分之五。
Step three, flip it and change the sign, so the perpendicular gradient is six over five.
第三步，翻转并变号，所以垂直方向的斜率是五分之六。
Step four, put the midpoint in to find c, which comes out as minus three fifths.
第四步，把中点代进去求 c，结果是负五分之三。
So the bisector is y equals six over five x, minus three fifths.
所以这条垂直平分线是 y 等于五分之六 x，减五分之三。
Three marks students give away.
三个学生常送掉的分。
First, when you find a gradient from two points, subtract in the same order on the top and on the bottom — swap one of them and the sign flips.
第一，用两个点求斜率时，上下要按同样的顺序相减—— 交换其中一个，符号就反了。
Second, perpendicular is not just a minus sign: you must flip the fraction as well as change the sign.
第二，垂直不只是加个负号： 你必须把分数翻过来，同时变号。
Third, the midpoint is an average, so add and halve, while the length uses Pythagoras.
第三，中点是平均值，所以相加再除以二， 而长度用的是勾股定理。
Get those three right and Topic Three is yours.
这三点做对，第三个专题就是你的了。
Four quick questions before you go.
走之前来四个小问题。
What is the gradient in y equals four x minus one?
y 等于四 x 减一里，斜率是多少？
It is four — read it straight off.
是四——直接读出来。
What is the gradient through one, two and four, eleven?
经过一、二和四、十一的直线，斜率是多少？
Three.
三。
What is the midpoint of one, two and four, six?
一、二和四、六的中点是多少？
Two point five, four.
二点五、四。
And what do perpendicular gradients multiply to?
垂直的两个斜率相乘等于多少？
Minus one.
负一。