当前位置:文档之家› (1502)黄金分割专项练习30题(有答案)

(1502)黄金分割专项练习30题(有答案)

(1502)黄金分割专项练习30题(有答案)
(1502)黄金分割专项练习30题(有答案)

黄金分割专项练习30题(有答案)

1,点C 在线段AB 上,若满足AC 2=BC?AB ,则称点C 为线段AB 的黄金分割点.如图 2, △ ABC ,/ A=36 ° BD 平分 / ABC 交 AC 于点 D .

D 是线段AC 的黄金分割点;

AD 的长.

3. 定义:如图1,点C 在线段 如图 2, △ ABC 中,AB=AC=2 (1) 求证:点D 是线段AC 的黄

金分割点;

C t

3

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2.如图,用长为

40cm 的细铁丝围成一个矩形 ABCD (AB > AD ). 1 .定义:如图 中,AB=AC=1 (1) 求证:点 (2) 求出线段 (1) 若这个矩形的面积等于

(2) 这个矩形的面积可能等于

(3)若这个矩形为黄金矩形( 99cm 2,求AB 的长度; 101cm 2吗?若能,求出 AB 的长度,若不能,说明理由; AD 与AB 之比等于黄金比 伍 ~ ' 2 ),求该矩形的面积.(结果保留根号)

AB 上,若满足AC 2=BC?AB ,则称点C 为线段AB 的黄金分割点. ,/

A=36 ° BD 平分 / ABC 交 AC 于点 D .

e

(2)求出线段AD的长.

4.作一个等腰三角形,使得腰与底之比为黄金比.

(1)尺规作图并保留作图痕迹;

(2)写出你的作法;

(3)证明:腰与底之比为黄金比.

5.(1)已知线段AB的长为2, P是AB的黄金分割点,求AP的长;

(2)求作线段AB的黄金分割点P,要求尺规作图,且使AP >PB .

』 --------------------- E

6.如图,线段AB的长度为1 .

(1)线段AB上的点C满足系式AC2=BC?AB,求线段AC的长度;

(选做)(2)线段AC上的点D满足关系式AD2=CD?AC,求线段AD的长度;

(选做)(3)线段AD上的点E满足关系式AE2=DE?AD,求线段AE的长度;上面各题的结果反映了什么规律?(提示:在每一小题中设x和I)

7.如图,在△ ABC中,AB=AC , / A=36 ° / 1 = / 2,请问点D是不是线段AC的黄金分割点.请说明理由.

&在^ ABC 中,AB=AC=2 , BC={亏-1 , / A=36 ° BD 平分/ ABC ,交于AC 于D .试说明点 D 是线段AC 的黄 金分割点.

为黄金矩形ABCD ?请你证明黄金矩形是由一个正方形和一个更小的黄金矩形构成. D Q

10.如图,设 AB 是已知线段,在 AB 上作正方形 ABCD ;取AD 的中点E ,连接EB ;延长DA 至F ,使EF=EB ; 以线段AF 为边作正方形 AFGH ?则点H 是AB 的黄金分割点.

为什么说上述的方法作出的点 H 是这条线段的黄金分割点,你能说出其中的道理吗?请试一试,说一说.

E ______ JG

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11.如图,已知△ ABC 中,D 是 AC 边上一点,/ A=36 ° / C=72 ° / ADB=108 ° 求证:

(1) AD=BD=BC ;

(2) 点D 是线段AC 的黄金分割点.

9.在数学上称长与宽之比为黄金分割比的矩形为黄金矩形, 如在矩形ABCD 中,当扭

」+产£C 时,称矩形ABCD

15?人的肚脐是人的身高的黄金分割点,一般来讲,当肚脐到脚底的长度与身高的比为

0.618时,是比较好看的黄

金身段.一个身高1.70m 的人,他的肚脐到脚底的长度为多少时才是黄金身段(保留两位小数)? 16. 如图所示,以长为 2的定线段AB 为边作正方形 ABCD ,取AB 的中点P ,连接PD ,在BA 的延长线上取点 F , 使PF=PD ,以AF 为边作正方形 AMEF ,点M 在AD 上.

(1) 求AM , DM 的长;

(2) 点M 是AD 的黄金分割点吗?为什么? F

— r

— D

?/

a

12 .已知 AB=2 , 点C 是AB 的黄金分割线,点 D 在AB 上,且AD 2

=BD?AB ,求器的值. 13.如果一个矩形

AR A 斥-1 ABCD (AB < BC )中,竺 —— BC 2 -0.618,那么这个矩形称为黄金矩形, 黄金矩形给人以美感. 在 ABFE (如图),请问矩形ABFE 是否是黄金矩形?请说明你

14?五角星是我们常见的图形,如图所示,其中,点

的长.

C ,

D 分别是线段 AB 的黄金分割点,AB=20cm ,求EC+CD C

黄金矩形ABCD

的结论的正确性. A E

3 F

17. 如图,点P 是线段AB 的黄金分割点,且 AP >BP ,设以AP 为边长的正方形面积为 S 1,以PB 为宽和以AB 为 长的矩形面积为S 2,试比较S 1与S 2的大小.

Sr

3斥一 1 18?如图,在平行四边形 ABCD 中,E 为边AD 延长线上的一点,且 D 为AE 的黄金分割点,即ADJ

BE 交DC 于点F ,已知AB=V5 + 1,求CF 的长.

2

折叠方法进行折叠,折叠后再展开,可以得到一个正方形 ABEF 和一个矩形 EFDC ,那么EFDC 这个矩形还是黄金

矩形吗?若是,请根据图 2证明你的结论;若不是,请说明理由.

/A / / '

3

AE ,

19?图1是一张宽与长之比为

徨丄:1

的矩形纸片,我们称这样的矩形为黄金矩形?同学们都知道按图 2所示的

(ffl 2)

20.(如图 1), 点P 将线段AB 分成一条较小线段 AP 和一条较大线段BP ,如果

善兰?,那么称点P 为线段AB 的 br Ai? 黄金分割点,设 柱影k ,则k 就是黄金比,并且k 战618 .

BP

J

(ffll ) E C

S '團2尸

(1) 以图1中的AP 为底,BP 为腰得到等腰△ APB (如图2),等腰△APB 即为黄金三角形,黄金三角形的定义为: 满足簷諾备-0.618的等腰三角形是黄金三角形;类似地,请你给出黄金矩形的定义:

; (2) 如图1,设AB=1,请你说明为什么

(3) 由线段的黄金分割点联想到图形的

3),点P 是线段AB 的黄金分割点,那么直线 CP 是△ ABC 的黄金分割线吗?请说明理由;

(4) 图3中的△ ABC 的黄金分割线有几条?

21.

在人体躯干(脚底到肚脐的长度)与身高的比例上,肚脐是理想的黄金分割点,即比例越接近

0.618,越给人 以美感.张女士原来脚底到肚脐的长度与身高的比为

0.60,她的身高为1.60m ,她应该选择多高的高跟鞋穿上看起

来更美?(精确到十分位) 22. 已知线段AB ,按照如下的方法作图:以 AB 为边作正方形 ABCD ,取AD 的中点E ,连接EB ,延长DA 到F , AF 为边,作正

方形 AFGH ,那么点H 是线段AB 的黄金分割点吗?请说明理由.

23?如图,用纸折出黄金分割点:裁一张正方的纸片

ABCD ,先折出BC 的中点E ,再折出线段AE ,然后通过折叠 使EB 落到线段EA 上,折出点B 的新位置B',因而EB'=EB .类似地,在 AB 上折出点B 〃使AB ''=AB 这时B " 就是AB 的黄金分割点.请你证明这个结论. k 约为 0.618; 黄金分割线”,类似地给出 黄金分割线”的定义:直线I 将一个面积为S 的 图形分成面积为S i 和面积为S 2的两部分

(设S 1< S 2),如果 ,那么称直线I 为该图形的黄金分割线.(如图

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