中考专页:2017年中考数学复习之几何探索类问题例析

探索类问题是近几年中考命题的重点,不少省市还作为压轴的大题。笔者研究了各地中考试卷,对命题特点、解题方法做了一些探讨。本文以中考题为例说明之,供同学们学习时参考。 一、实验型探索题…

中考专页:2017年中考数学复习之几何探索类问题例析

中考专页:2017年中考数学复习之几何探索类问题例析

探索类问题是近几年中考命题的重点,不少省市还作为压轴的大题。笔者研究了各地中考试卷,对命题特点、解题方法做了一些探讨。本文以中考题为例说明之,供同学们学习时参考。

一、实验型探索题

例1.等腰三角形是我们熟悉的图形之一,下面介绍一种等分等腰三角形面积的方法:如图1,在△ABC中,AB=AC,把底边BC分成m等份,连接顶点A和底边BC各等分点的线段,即可把这个三角形的面积m等分。中考专页:2017年中考数学复习之几何探索类问题例析

问题提出:任意给定一个正n边形,你能把它的面积m等分吗?

探究与发现:为了解决这个问题,我们先从简单问题入手怎样从正三角形的中心(正多边形的各对称轴的交点,又称为正多边形的中心)引线段,才能将这个正三角形的面积m等分?

如果要把正三角形的面积4等分,我们可以先连接正三角形的中心和各顶点(如图2(1)),这些线段将这个三角形分成了3个全等的等腰三角形);再把所得到的每个等腰三角形的底边4等分,连接中心和各边等分点(如图2(2),这些线段把这个三角形分成了12个面积相等的小三角形);最后依次把相邻的3个小三角形拼合在一起(如图2(3)),这样就能把这个正三角形的面积4等分了。中考专页:2017年中考数学复习之几何探索类问题例析

(1)实验与验证:仿照上述方法,利用刻度尺在图3中画出一种将正三角形的面积5等分的示意图。

中考专页:2017年中考数学复习之几何探索类问题例析

(2)猜想与证明:怎样从正三角形的中心引线段,才能将这个正三角形的面积m等分?叙述你的分法并说明理由。

(3)拓展与延伸:怎样从正方形(如图4)的中心引线段,才能将这个正方形的面积m等分(叙述分法即可,不要求说明理由)?

中考专页:2017年中考数学复习之几何探索类问题例析

(4)问题解决:怎样从正n边形(如图5)的中心引线段,才能使这个正n边形的面积m等分?(叙述分法,不要求说明理由)

分析:这类问题的特点是先给出一个解决问题的范例,然后要求解答一个类似的问题,最后将结论或方法推广到一般情况。这类问题文字较多,首先应弄清楚哪些是范例,哪些是要求解答的问题,然后详细阅读范例,从中领会解决问题的方法,并能运用这个方法解决问题。

解:(1)先连接正三角形的中心和各顶点,再把正三角形各边分别5等分,连接中心和各分点,然后将每3个相邻的小三角形拼在一起,就可将正三角形的面积5等分了(图略)。

(2)先连接正三角形的中心和各顶点,再把正三角形各边分别m等分,连接中心和各个分点,然后把每3个相邻的小三角形拼合在一起,即可把这个正三角形的面积m等分了。

理由:每个小三角形的底和高都相等,因此它们的面积都相等,每3个拼合在一起的图形面积当然也都相等,即把正三角形的面积m等分。

(3)先连接正方形的中心和各顶点,然后将正方形各边m等分,连接中心和各分点,再依次将相邻的4个小三角形拼合在一起,这就把这个正方形的面积m等分了。

(4)连接正n边形的中心和各顶点,然后将这个正n边形各边m等分,再依次将n个相邻的小三角形拼在一起,这就将这个正n边形的面积m等分了。

中考专页:2017年中考数学复习之几何探索类问题例析

二、操作型探索题

例2.已知线段AC=8,BD=6。(1)已知线段AC⊥BD于O(O不与A、B、C、D四点重合),设图6(1)、图6(2)和图6(3)中的四边形ABCD的面积分别为S1、S2、S3,则S1=_________,S2=________,S3=_________;中考专页:2017年中考数学复习之几何探索类问题例析

(2)如图6(4),对于线段AC与线段BD垂直相交(垂足O不与点A、B、C、D重合)的任意情形,请你就四边形ABCD面积的大小提出猜想,并证明你的结论;

(3)当线段BD与AC(或CA)的延长线垂直相交时,猜想顺次连接点A、B、C、D所围成的封闭图形的面积是多少。

分析:题(1)实际上是将BD沿AC由下向上移动,计算BC在不同位置时四边形ABCD的面积,再观察计算结果。题(2)是AC沿BD左右移动,计算四边形ABCD的面积,再观察计算结果。题(3)是在更一般的情况下探索规律。这种由浅入深的探索方式是中考探索类问题的特点。

解:(1)24 24 24

(2)对于线段AC与线段BD垂直相交(垂足O不与点A、C、B、D重合)的任意情形,四边形ABCD的面积为定值24。证明如下:中考专页:2017年中考数学复习之几何探索类问题例析

(3)所围成的封闭图形的面积仍为24。

三、观察猜想型探索题

例3. 如图7,正方形ABCD的边CD在正方形EFGC的边CE上,连接BE、DG。

中考专页:2017年中考数学复习之几何探索类问题例析

(1)观察并猜想BE与DG之间的大小关系,并证明你的结论;

(2)图7中是否存在通过旋转能够互相重合的三角形?若存在,请说明旋转过程;若不存在,说明理由。

分析:证明题是直接给出结论,要求寻找结论成立的理由,而这一类探索题是题目没有给出结论,要求自己下结论,并证明结论成立。这就要求有较强的观察猜想能力。

解:(1)BE=DG,证明如下:

在Rt△BCE和Rt△DCG中,BC=CD,CE=CG,

∴△BCE≌△DCG。故BE=DG。

(2)将Rt△BCE绕点C顺时针旋转90°,可与Rt△DCG重合。

四、图形计数型探索题

例4.如图8,在图(1)中,互不重叠的三角形有4个,在图(2)中,互不重叠的三角形有7个,在图(3)中,互不重叠的三角形有10个,…,则在图(n)中互不重叠的三角形有_______个(用含n的代数式表示)中考专页:2017年中考数学复习之几何探索类问题例析

分析:这类图形计数型探索题有线段计数、射线计数、角计数等。解这类题首先要通过几个具体图形寻找规律,然后写出公式,或称一般表达式。解题的关键是找规律。

解:图(1):1+1×3=4;图(2):1+2×3=7;图(3):1+3×3=10。

所以图(n)中有1+3n个互不重叠的三角形,应填3n+1。

五、其他类型探索题

例5.如图9,已知AC、AB是⊙O的弦,AB>AC。

中考专页:2017年中考数学复习之几何探索类问题例析

中考专页:2017年中考数学复习之几何探索类问题例析

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