当前位置：
＞＞＞图，在矩形ABCD中，AB=3，BC=2，以BC为直径在矩形内作半圆，自点..
图，在矩形ABCD中，AB=3，BC=2，以BC为直径在矩形内作半圆，自点A作半圆的切线AE，则sin∠CBE=（　　）A．63B．23C．13D．1010
题型：单选题难度：中档来源：杭州一模
取BC的中点O，则O为圆心，连接OE，AO，AO与BE的交点是F∵AB，AE都为圆的切线∴AE=AB∵OB=OE，AO=AO∴△ABO≌△AEO（SSS）∴∠OAB=∠OAE∴AO⊥BE在直角△AOB里AO2=OB2+AB2∵OB=1，AB=3∴AO=10易证明△BOF∽△AOB∴BO：AO=OF：OB∴1：10=OF：1∴OF=1010sin∠CBE=OFOB=1010故选D．
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据魔方格专家权威分析，试题“图，在矩形ABCD中，AB=3，BC=2，以BC为直径在矩形内作半圆，自点..”主要考查你对&&直线与圆的位置关系（直线与圆的相交，直线与圆的相切，直线与圆的相离），锐角三角函数的定义&&等考点的理解。关于这些考点的“档案”如下：
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因为篇幅有限，只列出部分考点，详细请访问。
直线与圆的位置关系（直线与圆的相交，直线与圆的相切，直线与圆的相离）锐角三角函数的定义
直线与圆的位置关系:直线与圆的位置关系有三种：直线与圆相交，直线与圆相切，直线与圆相离。 （1）相交：直线和圆有两个公共点时，叫做直线和圆相交，这时直线叫做圆的割线，公共点叫做交点AB与⊙O相交，d&r； （2）相切：直线和圆有唯一公共点时，叫做直线和圆相切，这时直线叫做圆的切线，这个唯一的公共点叫做切点。AB与⊙O相切，d=r。（3）相离：直线和圆没有公共点时，叫做直线和圆相离,AB与圆O相离，d&r。（d为圆心到直线的距离）直线与圆的三种位置关系的判定与性质： （1）数量法：通过比较圆心O到直线距离d与圆半径的大小关系来判定， 如果⊙O的半径为r，圆心O到直线l的距离为d，则有： 直线l与⊙O相交d&r； 直线l与⊙O相切d=r； 直线l与⊙O相离d&r； （2）公共点法：通过确定直线与圆的公共点个数来判定。 直线l与⊙O相交d&r2个公共点； 直线l与⊙O相切d=r有唯一公共点； 直线l与⊙O相离d&r无公共点 。圆的切线的判定和性质&&& （1）切线的判定定理：经过半径的外端并且垂直于这条半径的直线是圆的切线。 （2）切线的性质定理：圆的切线垂直于经过切点的半径。 切线长：在经过圆外一点的圆的切线上，这点和切点之间的线段的长叫做这点到圆的切线长。 切线长定理：从圆外一点引圆的两条切线，它们的切线长相等，圆心和这一点的连线平分两条切线的夹角。 直线与圆的位置关系判定方法:平面内，直线Ax+By+C=0与圆x2+y2+Dx+Ey+F=0的位置关系判断一般方法是：1.由Ax+By+C=0，可得y=（-C-Ax）/B，（其中B不等于0），代入x2+y2+Dx+Ey+F=0，即成为一个关于x的方程如果b2-4ac&0，则圆与直线有2交点，即圆与直线相交。如果b2-4ac=0，则圆与直线有1交点，即圆与直线相切。如果b2-4ac&0，则圆与直线有0交点，即圆与直线相离。2.如果B=0即直线为Ax+C=0，即x=-C/A，它平行于y轴（或垂直于x轴），将x2+y2+Dx+Ey+F=0化为（x-a）2+（y-b）2=r2。令y=b，求出此时的两个x值x1、x2，并且规定x1&x2，那么：& 当x=-C/A&x1或x=-C/A&x2时，直线与圆相离；当x1&x=-C/A&x2时，直线与圆相交。&锐角三角函数：锐角角A的正弦（sin）,余弦（cos）和正切（tan）,余切（cot）以及正割（sec），余割（csc）都叫做角A的锐角三角函数。初中学习的 锐角三角函数值的定义方法是在直角三角形中定义的，所以在初中阶段求锐角的三角函数值，都是通过构造直角三角形来完成的，即把这个角放到某个直角三角形中。所谓锐角三角函数是指：我们初中研究的都是锐角的三角函数。初中研究的锐角的三角函数为：正弦（sin），余弦（cos），正切（tan）。正弦：在直角三角形中，锐角A的对边与斜边的比叫做∠A的正弦，记作sinA，即；余弦：在直角三角形中，锐角A的邻边与斜边的比叫做∠A的余弦，记作cosA，即；正切：在直角三角形中，锐角A的对边与邻边的比叫做∠A的正切，记作tanA，即，锐角A的正弦、余弦、正切都叫做A的锐角三角函数。锐角三角函数的增减性：1.锐角三角函数值都是正值2.当角度在0°～90°间变化时，正弦值随着角度的增大（或减小）而增大（或减小） ，余弦值随着角度的增大（或减小）而减小（或增大） ；正切值随着角度的增大（或减小）而增大（或减小） ，余切值随着角度的增大（或减小）而减小（或增大）；正割值随着角度的增大（或减小）而增大（或减小），余割值随着角度的增大（或减小）而减小（或增大）。3.当角度在0°≤A≤90°间变化时，0≤sinA≤1, 1≥cosA≥0；当角度在0°&A0, cotA&0。锐角三角函数的关系式：同角三角函数基本关系式tanα·cotα=1sin2α·cos2α=1cos2α·sin2α=1sinα/cosα=tanα=secα/cscαcosα/sinα=cotα=cscα/secα(sinα)2+(cosα)2=11+tanα=secα1+cotα=cscα诱导公式sin（－α）=－sinαcos（－α）=cosαtan（－α）=－tanαcot（－α）=－cotαsin（π/2－α）=cosαcos（π/2－α）=sinαtan（π/2－α）=cotαcot（π/2－α）=tanαsin（π/2+α）=cosαcos（π/2+α）=－sinαtan（π/2+α）=－cotαcot（π/2+α）=－tanαsin（π－α）=sinαcos（π－α）=－cosαtan（π－α）=－tanαcot（π－α）=－cotαsin（π+α）=－sinαcos（π+α）=－cosαtan（π+α）=tanαcot（π+α）=cotαsin（3π/2－α）=－cosαcos（3π/2－α）=－sinαtan（3π/2－α）=cotαcot（3π/2－α）=tanαsin（3π/2+α）=－cosαcos（3π/2+α）=sinαtan（3π/2+α）=－cotαcot（3π/2+α）=－tanαsin（2π－α）=－sinαcos（2π－α）=cosαtan（2π－α）=－tanαcot（2π－α）=－cotαsin（2kπ+α）=sinαcos（2kπ+α）=cosαtan（2kπ+α）=tanαcot（2kπ+α）=cotα(其中k∈Z)二倍角、三倍角的正弦、余弦和正切公式Sin(2α)=2sinαcosαCos(2α)=（cosα）2－(sinα)2=2(cosα)2－1=1－2(sinα)2Tan(2α)=2tanα/(1-tanα)sin(3α)=3sinα-4sin3α=4sinα·sin(60°+α)sin(60°-α)cos(3α)=4cos3α-3cosα=4cosα·cos(60°+α)cos(60°-α)tan(3α)=(3tanα-tan3α)/(1-3tan2α)=tanαtan(π/3+α)tan(π/3-α)和差化积、积化和差公式sinα+sinβ=2sin[(α+β)/2]·cos[(α-β)/2]sinα-sinβ=2cos[(α+β)/2]·sin[(α-β)/2]cosα+cosβ=2cos[(α+β)/2]·cos[(α-β)/2]cosα-cosβ=-2sin[(α+β)/2]·sin[(α-β)/2]sinαcosβ=-[sin（α+β）+sin（α－β）]sinαsinβ=-[1][cos（α+β）-cos（α-β）]/2cosαcosβ=[cos（α+β）+cos（α-β）]/2sinαcosβ=[sin（α+β）+sin（α-β）]/2cosαsinβ=[sin（α+β）-sin（α-β）]/2
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13133592603692648021318093045283259From Wikipedia, the free encyclopedia
For the fictional character Julie Walters, see .
Julia Mary "Julie" Walters,
(born 22 February 1950) is an English actress and writer. She has won two , four
and received the
Walters first came to international prominence in 1983, for playing the title role in . It was a role she had created on the West End stage and it earned her an
nomination for . It also won her a
and a . She received a second Academy Award nomination, this time for , for her role in the 2000 film , which also won her a BAFTA. Her other film roles include
(2003) and
(2008). She has also played
in seven of the eight
films (). On stage, she won an
for the 2001 production of .
On television, she is well known for her collaborations with
and has appeared with her in several television shows including
(1994) and
(). She has won the
four times, for My Beautiful Son (2001), Murder (2002),
(2003) and as
(2010). She also starred in
in 2009, which won her an
for Best Actress. In 2006, she came fourth in 's poll of the public's
in Britain.
Walters was born in St. Chad's Hospital, , , , then the maternity hospital for , . Her parents, Mary Bridget (née O'Brien), an Irish Catholic postal clerk born in Ireland, and Thomas Walters, an English builder and decorator, lived at 69 Bishopton Road, near Lightwoods Park, in the
area of Smethwick. The youngest of five children and the third to survive birth, Walters had an early education at a convent school and later at
on Holly Lane in Smethwick, although she was asked to leave at the end of her lower sixth because of her "high jinks". In an interview with Alison Oddey, Walters said about her early schooling: "I was never going to be academic, so [my mother] suggested that I try teaching or nursing [...] I'd been asked to leave school, so I thought I'd better do it."
Her first job was in insurance at the age of 15. At 18 she trained as a nurse at the , Birmingham, and worked on the ophthalmic, casualty and coronary care wards during the 18 months she spent there. Walters decided to leave nursing, and studied English and drama at Manchester Polytechnic (now ) and pursued a career in the . Walters worked for the
in Liverpool in the mid-1970s, alongside several other notable performers: , , ,
Walters first received notice as the occasional partner of comedienne , whom she had briefly met in . The two first worked together in the 1978 theatre revue , followed by the television adaptation of Wood's play . They went on to appear in their own
series, , in 1982. They have continued to perform together frequently over the years. The -winning BBC follow-up, , featured one of Walters's best-known roles, Mrs. Overall in Wood's
soap opera,
(she later appeared in , and received an
nomination for her efforts).
Before making her London stage debut in Educating Rita, Walters had worked in regional theatre, stand-up comedy and cabaret. Her first serious acting role on TV was in the classic
in 1982, and she broke into films with her -nominated, BAFTA Best Actress award-winning and
Best Actress – Motion Picture Musical/Comedy award-winning performance opposite
(1983), a role she had created on the West End stage.
In 1985, she played 's mother Pauline in the TV adaptation of . Walters appeared in the lead role of
in the 1987 film
– a dramatic comedy about a British brothel owner. Then she played the lead character's wife, June, in the film , released in 1988. She also appeared as Mrs. Peachum in the 1989 film version of , which was renamed
for the screen.
In 1991, Walters starred opposite
and had a one-off television special, , which featured writing contributions from
and . In 1998 she starred as the Fairy Godmother in the ITV pantomime – , alongside actors , , ,
and . The show was first broadcast 25 December 1998 on ITV1 and continues to be shown every year around Christmas on ITV2.
Walters has won numerous other acting awards, and was appointed officer of the
(OBE) in 1999 and raised to commander level (CBE) in the
for her services to drama. In 2001 she won a
for her performance in 's . She received her second Oscar nomination and won a BAFTA for her supporting role as the ballet teacher in
(2000). In 2002 she again won a BAFTA for her performance as 's mother in My Beautiful Son.
Walters also played
(2009) and
(2010) and
(2011) and got international success with it.
is the only
film to have not starred Walters.
In 2003 Walters starred as a widow (Annie Clark) determined to make some good come out of her husband's death from cancer in , which also starred
and . In 2005 Walters again starred as an inspirational real-life figure,
drama Ahead of the Class.
In 2006, she came fourth in ITV's poll of the public's 50 Greatest Stars, coming four places above frequent co-star . Also in 2006, she starred in the film
(who played her son Ron in the
series), and later had a leading role in the 's adaptation of 's novel .
In the summer of 2006, Walters published her first novel, Maggie's Tree. The novel, concerning a group of English actors in Manhattan and published by , was described as "a disturbing and thought-provoking novel about mental torment and the often blackly comic, mixed-up ways we view ourselves and misread each other." Another reviewer described the novel as "the work of a writer who knows what she's doing. There's nothing tentative about the writing, and Walters brings her experiences as an actress to bear on the page. ... you do have the sensation of entering someone else's mind and of looking through someone else's eyes."
Walters starred in 's Christmas 2007 TV advertising campaign. She also appeared alongside
in UK Nintendo DS Brain Training television advertisements, and in a
about smoke alarms. In summer 2008, Walters appeared in the film version of , playing Rosie Mulligan, marking her second high profile musical, after .
Walters played
in the BBC Drama , an adaptation of the real-life story of Mrs. Whitehouse who campaigned for "taste and decency on television". Walters commented, "I am very excited to be playing Mary Whitehouse, and to be looking at the time when she attacked the BBC and started to make her name." Filth won Best Motion Picture Made for Television, and Walters was nominated for Best Actress in a Miniseries or a Motion Picture Made For Television, at the 2008 13th Annual Satellite Awards.[]
In 2009 Walters received a star in the
on Birmingham's Golden Mile, . She said: "I am very honoured and happy that the people of Birmingham and the West Midlands want to include me in their Walk of Stars and I look forward to receiving my star. Birmingham and the West Midlands is where I' these are my roots and in essence it has played a big part in making me the person I am today". Her other awards include an International Emmy with Ben Whishaw for . She also appeared as Petula Gordino in Wood's sitcom .
Walters played the late MP and
in a drama for
broadcast in early 2010. She had misgivings about taking on the role because of the differences in their physical appearance, but the result was highly praised by critics.
In July 2012 Walters appeared in the
production
as Mistress Quickly in Shakespeare's .
In 2012 she worked with
to promote one of their life insurance products targeted at people over 50. Walters was seen in television advertisements, at
website and in other marketing material helping to raise awareness for life insurance.
Walters appeared in The Last of the Haussmans at the
in June 2012. The production was broadcast to cinemas around the world through the
programme.
Walters' relationship with Grant Roffey, an
patrol man, began after a whirlwind romance. The couple have a daughter, Maisie Mae Roffey (born 26 April 1988, , London), but did not marry until 1997, 11 years into their relationship, when they went to New York City. The couple live on an organic farm run by Roffey near .
In August 2014 she featured in the first episode in the eleventh series of the BBC genealogy series . The programme revealed that her maternal ancestors played an active part in the struggle for more rights for Irish tenant farmers, known as the , which started in 1879. Although not included in the programme, Walters' paternal grandfather, Thomas Walters, was a veteran of the . He was killed in action in
in June 1915, serving with the 2nd Battalion of the
and is commemorated at the , France.
TV: 1 episode
girl in surgery
TV: 1 episode
woman in waiting room
Debbie/Valerie
TV: 2 episodes
Jean Watson
TV: 2 episodes
Julie Stephens
Screenplay
Frances/Julie
TV: 3 episodes
Julie Stephens
various roles
Mrs Morgan
TV: 1 episode
Angie Todd
TV: 1 episode
June Potter
TV: 1 episode
Susan "Rita" White
TV: 1 episode
Pauline Mole
TV: 5 episodes
Car Trouble
Jacqueline Spong
various characters
TV: 13 episodes
Christina Painter
Elsie Orton
TV: 1 episode
Overall, Mrs.Mrs. Overall
TV: 6 episodes
TV: 1 episode: "Her Big Chance"
June Edwards
Peachum, MrsMrs Peachum
various roles
TV: 3 episodes
Killing Dad or How to Love Your Mother
Julie Walters and Friends
herself/various roles
Murray, MrsMrs Murray
TV: 7 episodes
various roles
Mavis / Monica
TV: 2 episodes
(aka The Wedding Gift)
Diana Longden
TV: 1 episode
Madame Danzard
Mrs Capstan
Jake's Progress
Julie Diadoni
TV: 6 episodes
Little Red Riding Hood / Grandma
Marjorie Beasley
Maureen Hardcastle
Miss Gideon
Paula Hepburn
TV: 5 episodes
Fairy Godmother
Jackie Simpson
Bernie McPhelimy
TV: 1 episode: "The Outside Dog"
TV: 9 episodes
Mann, MrsMrs Mann
TV: 4 episodes
Wilkinson, MrsMrs Wilkinson
All Forgotten
Zasyekin, PrincessPrincess Zasyekin
My Beautiful Son
Sheila Fitzpatrick
Mrs. Weasley
Angela Maurer
TV: 4 episodes
Mrs. Weasley
The Return
Lizzie Hunt
Mrs. Weasley
Mickybo's Ma
Gwen Traherne
Evie Walton
Holland, MrsMrs Holland
Mrs. Weasley
Austen, MrsMrs Austen
Mrs. Weasley
Anne Turner, DrDr Anne Turner
Bo Beaumont/Mrs. Overall
Mrs. Weasley
Montague, MissMiss Montague
Mrs. Weasley
Emma Watts
Mistress Quickly
Thread of Evidence
Betty Beesom
Margaret Cox Ruskin
Yvonne Potts
Harry's Nan
(London debut) Irene Tinsley, Funny Peculiar, Mermaid Theatre, then Garrick Theatre, London, 1976
Vera, Breezeblock Park, Mermaid Theatre, then Whitehall Theatre, London, 1977
Irene Goodnight, Flaming Bodies, ICA Theatre, London, 1979
Rita, Educating Rita, Royal Shakespeare Company, London, 1980
Having a Ball, Lyric Hammersmith Theatre, London, 1981
Dotty, , , 1984
Fool for Love, Royal National Theatre, London, 1984–85
Macbeth, Leicester Haymarket Theatre, 1985
When I Was a Girl I Used to Scream and Shout, Whitehall Theatre, 1986
Frankie and Johnny in the Claire de Lune, Comedy Theatre, 1989
Serafina, The Rose Tattoo, Playhouse, London, 1991
All My Sons, Royal National Theatre, 2000
Acorn Antiques: The Musical, 2005
Also appeared in The Taming of the Shrew, produced in Liverpool, E and in Jumpers, Royal E performed with *Everyman Theatre, Liverpool and Bristol Old Vic.
The Last of the Haussmans, Royal National Theatre, London, 2012
Star at the Birmingham Walk of Stars
Educating Rita
Rita Susan White
Nominated –
Boys from the Black Stuff
Angie Todd
Nominated –
Personal Services
Christine Painter
Nominated –
Stepping Out
Nominated –
Billy Elliot
Sandra Wilkinson
Nominated –
Nominated –
Nominated –
All My Sons
Kate Keller
My Beautiful Son
Sheila Fitzpatrick
Angela Maurer
Canterbury Tales
Driving Lessons
Evie Walton
Silver George for Best Actress ()
A Short Stay in Switzerland
Dr Anne Turner
Award for Best Actress
Award for Best Actress
Julie Walters has won eight BAFTAs, six competitive awards plus two honorary awards. The first honorary award was a special BAFTA that she received at a tribute evening in 2003, before receiving the
In 2000, she was awarded the Dilys Powell Award for Excellence in Film by the UK Critics' Circle.
She became an
in 1999 and CBE in 2008.
(Weidenfeld & Nicolson, 2007)
(Orion Books, 2009)
Walters, Julie (2008). That's Another Story: The Autobiography. Weidenfeld & Nicolson, London. p. 2.  .
Scott, Danny (3 September 2006). . The Times (London) 2010.
Mottram, James (14 May 2001). . The Guardian (London) 2010.
Walters, Julie (2008). That's Another Story: The Autobiography. Orion Publishing Co. p. 1.  .
Performing Women: Stand-ups, Strumpets and Itinerants, by Alison Oddey, Palgrave Macmillan, 2005. p. 305
Walters, Julie (2008). That's Another Story: The Autobiography. Orion Publishing Co. p. 100.  .
Walters, Julie (2008). That's Another Story: The Autobiography. Orion Publishing Co. pp. 102–123.  .
Nigel Farndale (25 March 2009). . The Daily Telegraph (UK).
Saner, Emine (13 October 2006). . The Guardian (London) 2010.
, . The Guardian, 14 October 2006. Retrieved 2 September 2013.
, . The Independent, 13 October 2006. Retrieved 2 September 2013.
. Daily Mail (UK). 4 June 2009.
. BBC. 20 January 2010.
James Rampton (29 January 2010). . The Independent (UK).
(Press release). BBC Drama Publicity. 24 November 2011. Archived from
on 30 December .
. National Theatre 2012.
. MIFF 2013.
– interactive video interview presented by BFI Screenonline and British Telecom
: Hidden categories:直角三角形纸片的两个直角边分别为6,8，现将三角形ABC如图那样折叠，使点A与点B重合，折痕为DE，则tan角CBE的值为
直角三角形纸片的两个直角边分别为6,8，现将三角形ABC如图那样折叠，使点A与点B重合，折痕为DE，则tan角CBE的值为 5
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当前分类官方群专业解答学科习题，随时随地的答疑辅导当前位置：
＞＞＞如图甲，四边形OABC的边OA、OC分别在x轴、y轴的正半轴上，顶点在..
如图甲，四边形OABC的边OA、OC分别在x轴、y轴的正半轴上，顶点在B点的抛物线交x轴于点A、D，交y轴于点E，连接AB、AE、BE．已知tan∠CBE=，A（3，0），D（﹣1，0），E（0，3）．（1）求抛物线的解析式及顶点B的坐标；（2）求证：CB是△ABE外接圆的切线；（3）试探究坐标轴上是否存在一点P，使以D、E、P为顶点的三角形与△ABE相似，若存在，直接写出点P的坐标；若不存在，请说明理由；（4）设△AOE沿x轴正方向平移t个单位长度（0＜t≦3）时，△AOE与△ABE重叠部分的面积为s，求s与t之间的函数关系式，并指出t的取值范围．
题型：解答题难度：偏难来源：湖北省中考真题
解：由题意，设抛物线解析式为y=a（x﹣3）（x+1）．将E（0，3）代入上式，解得：a=﹣1．∴y=﹣x2+2x+3．则点B（1，4）．（2）证明：如图1，过点B作BM⊥y于点M，则M（0，4）．在Rt△AOE中，OA=OE=3，∴∠1=∠2=45°，AE==3．在Rt△EMB中，EM=OM﹣OE=1=BM，∴∠MEB=∠MBE=45°，BE==．∴∠BEA=180°﹣∠1﹣∠MEB=90°．∵AB是△ABE外接圆的直径．在Rt△ABE中，tan∠BAE===tan∠CBE，∴∠BAE=∠CBE．在Rt△ABE中，∠BAE+∠3=90°，∴∠CBE+∠3=90°．∴∠CBA=90°，即CB⊥AB．∴CB是△ABE外接圆的切线．（3）解：Rt△ABE中，∠AEB=90°，tan∠BAE=，sin∠BAE=，cos∠BAE=；若以D、E、P为顶点的三角形与△ABE相似，则△DEP必为直角三角形；①DE为斜边时，P1在x轴上，此时P1与O重合；由D（﹣1，0）、E（0，3），得OD=1、OE=3，即tan∠DEO==tan∠BAE，即∠DEO=∠BAE满足△DEO∽△BAE的条件，因此 O点是符合条件的P1点，坐标为（0，0）．②DE为短直角边时，P2在x轴上；若以D、E、P为顶点的三角形与△ABE相似，则∠DEP2=∠AEB=90°，sin∠DP2E=sin∠BAE=；而DE==，则DP2=DE×sin∠DP2E=×=10，OP2=DP2﹣OD=9即：P2（9，0）；③DE为长直角边时，点P3在y轴上；若以D、E、P为顶点的三角形与△ABE相似，则∠EDP3=∠AEB=90°，cos∠DEP3=cos∠BAE=；则EP3=DE×cos∠DEP3=×=，OP3=EP3﹣OE=；综上，得：P1（0，0），P2（9，0），P3（0，﹣）．（4）解：设直线AB的解析式为y=kx+b．将A（3，0），B（1，4）代入，得解得∴y=﹣2x+6．过点E作射线EF∥x轴交AB于点F，当y=3时，得x=，∴F（，3）．情况一：如图2，当0＜t≦时，设△AOE平移到△DNM的位置，MD交AB于点H，MN交AE于点G．则ON=AD=t，过点H作LK⊥x轴于点K，交EF于点L．由△AHD∽△FHM，得，即．解得HK=2t．∴S阴=S△MND﹣S△GNA﹣S△HAD=×3×3﹣（3﹣t）2﹣t×2t=﹣t2+3t．情况二：如图3，当＜t≦3时，设△AOE平移到△PQR的位置，PQ交AB于点I，交AE于点V．由△IQA∽△IPF，得．即，解得IQ=2（3﹣t）．∴S阴=S△IQA﹣S△VQA=×（3﹣t）×2（3﹣t）﹣（3﹣t）×2=（3﹣t）2=t2﹣3t+．综上所述：s=．
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据魔方格专家权威分析，试题“如图甲，四边形OABC的边OA、OC分别在x轴、y轴的正半轴上，顶点在..”主要考查你对&&求二次函数的解析式及二次函数的应用，直线与圆的位置关系（直线与圆的相交，直线与圆的相切，直线与圆的相离），相似三角形的性质&&等考点的理解。关于这些考点的“档案”如下：
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求二次函数的解析式及二次函数的应用直线与圆的位置关系（直线与圆的相交，直线与圆的相切，直线与圆的相离）相似三角形的性质
求二次函数的解析式：最常用的方法是待定系数法，根据题目的特点，选择恰当的形式，一般，有如下几种情况： （1）已知抛物线上三点的坐标，一般选用一般式； （2）已知抛物线顶点或对称轴或最大（小）值，一般选用顶点式； （3）已知抛物线与x轴的两个交点的横坐标，一般选用两点式； （4）已知抛物线上纵坐标相同的两点，常选用顶点式。 二次函数的应用：（1）应用二次函数才解决实际问题的一般思路： 理解题意；建立数学模型；解决题目提出的问题。 （2）应用二次函数求实际问题中的最值： 即解二次函数最值应用题，设法把关于最值的实际问题转化为二次函数的最值问题，然后按求二次函数最值的方法求解。求最值时，要注意求得答案要符合实际问题。 二次函数的三种表达形式：①一般式：y=ax2+bx+c(a≠0,a、b、c为常数)，顶点坐标为 [,]把三个点代入函数解析式得出一个三元一次方程组，就能解出a、b、c的值。
②顶点式：y=a(x-h)2+k(a≠0,a、h、k为常数),顶点坐标为对称轴为直线x=h，顶点的位置特征和图像的开口方向与函数y=ax2的图像相同，当x=h时，y最值=k。有时题目会指出让你用配方法把一般式化成顶点式。例：已知二次函数y的顶点(1,2)和另一任意点(3,10)，求y的解析式。解：设y=a(x-1)2+2，把(3,10)代入上式，解得y=2(x-1)2+2。注意：与点在平面直角坐标系中的平移不同，二次函数平移后的顶点式中，h&0时，h越大，图像的对称轴离y轴越远，且在x轴正方向上，不能因h前是负号就简单地认为是向左平移。具体可分为下面几种情况：当h&0时，y=a(x-h)2的图象可由抛物线y=ax2向右平行移动h个单位得到；当h&0时，y=a(x-h)2的图象可由抛物线y=ax2向左平行移动|h|个单位得到；当h&0,k&0时，将抛物线y=ax2向右平行移动h个单位，再向上移动k个单位，就可以得到y=a(x-h)2+k的图象；当h&0,k&0时，将抛物线y=ax2向右平行移动h个单位，再向下移动|k|个单位可得到y=a(x-h)2+k的图象；当h&0,k&0时，将抛物线y=ax2向左平行移动|h|个单位，再向上移动k个单位可得到y=a(x-h)2+k的图象；当h&0,k&0时，将抛物线y=ax2向左平行移动|h|个单位，再向下移动|k|个单位可得到y=a(x-h)2+k的图象。
③交点式：y=a(x-x1)(x-x2) (a≠0) [仅限于与x轴即y=0有交点时的抛物线，即b2-4ac≥0] .已知抛物线与x轴即y=0有交点A（x1，0）和 B（x2，0）,我们可设y=a(x-x1)(x-x2),然后把第三点代入x、y中便可求出a。由一般式变为交点式的步骤：二次函数∵x1+x2=-b/a， x1?x2=c/a(由韦达定理得),∴y=ax2+bx+c=a(x2+b/ax+c/a)=a[x2-(x1+x2)x+x1?x2]=a(x-x1)(x-x2).重要概念：a，b，c为常数，a≠0，且a决定函数的开口方向。a&0时，开口方向向上；a&0时，开口方向向下。a的绝对值可以决定开口大小。a的绝对值越大开口就越小，a的绝对值越小开口就越大。能灵活运用这三种方式求二次函数的解析式；能熟练地运用二次函数在几何领域中的应用；能熟练地运用二次函数解决实际问题。二次函数的其他表达形式：①牛顿插值公式:f(x)=f[x0]+f[x0,x1](x-x0)+f[x0,x1,x2](x-x0)(x-x1)+...f[x0,...xn](x-x0)...(x-xn-1)+Rn(x)由此可引导出交点式的系数a=y/(x·x)(y为截距)　二次函数表达式的右边通常为二次三项式。双根式y=a(x-x1)*(x-x2)若ax2+bx+c=0有两个实根x1,x2，则y=a(x-x1)(x-x2)此抛物线的对称轴为直线x=(x1+x2)/2。③三点式已知二次函数上三个点，（x1,f(x1)）（x2,f(x2)）（x3,f(x3)）则f(x)=f(x3)(x-x1)(x-x2)/(x3-x1)(x3-x2)+f(x2)(x-x1)*(x-x3)/(x2-x1)(x2-x3)+f(x1)(x-x2)(x-x3)/(x1-x2)(x1-x3)与X轴交点的情况当△=b2-4ac&0时，函数图像与x轴有两个交点。(x1,0), (x2,0);当△=b2-4ac=0时，函数图像与x轴只有一个交点。(-b/2a，0)。Δ=b2-4ac&0时，抛物线与x轴没有交点。X的取值是虚数（x=-b±√b2－4ac的值的相反数，乘上虚数i，整个式子除以2a）二次函数解释式的求法：就一般式y=ax2＋bx＋c（其中a，b，c为常数，且a≠0）而言，其中含有三个待定的系数a ，b ，c．求二次函数的一般式时，必须要有三个独立的定量条件，来建立关于a ，b ，c 的方程，联立求解，再把求出的a ，b ，c 的值反代回原函数解析式，即可得到所求的二次函数解析式。
1.巧取交点式法：知识归纳：二次函数交点式：y＝a(x－x1)(x－x2) （a≠0）x1，x2分别是抛物线与x轴两个交点的横坐标。已知抛物线与x轴两个交点的横坐标求二次函数解析式时，用交点式比较简便。①典型例题一：告诉抛物线与x轴的两个交点的横坐标，和第三个点，可求出函数的交点式。例：已知抛物线与x轴交点的横坐标为-2和1 ，且通过点（2，8），求二次函数的解析式。点拨：解设函数的解析式为y＝a(x+2)(x-1)，∵过点（2，8），∴8＝a(2+2)(2-1)。解得a=2，∴抛物线的解析式为：y＝2(x+2)(x-1)，即y＝2x2+2x-4。②典型例题二：告诉抛物线与x轴的两个交点之间的距离和对称轴，可利用抛物线的对称性求解。例：已知二次函数的顶点坐标为（3，-2），并且图象与x轴两交点间的距离为4，求二次函数的解析式。点拨：在已知抛物线与x轴两交点的距离和顶点坐标的情况下，问题比较容易解决．由顶点坐标为（3，-2）的条件，易知其对称轴为x＝3，再利用抛物线的对称性，可知图象与x轴两交点的坐标分别为（1，0）和（5，0）。此时，可使用二次函数的交点式，得出函数解析式。
2.巧用顶点式：顶点式y=a(x－h)2+k（a≠0），其中（h，k）是抛物线的顶点。当已知抛物线顶点坐标或对称轴，或能够先求出抛物线顶点时，设顶点式解题十分简洁，因为其中只有一个未知数a。在此类问题中，常和对称轴，最大值或最小值结合起来命题。在应用题中，涉及到桥拱、隧道、弹道曲线、投篮等问题时，一般用顶点式方便．①典型例题一：告诉顶点坐标和另一个点的坐标，直接可以解出函数顶点式。例：已知抛物线的顶点坐标为（-1，-2），且通过点（1，10），求此二次函数的解析式。点拨：解∵顶点坐标为（-1，-2），故设二次函数解析式为y=a(x+1)2-2 （a≠0）。把点（1，10）代入上式，得10=a·(1+1)2-2。∴a=3。∴二次函数的解析式为y=3(x+1)2-2，即y=3x2+6x+1。②典型例题二：如果a&0，那么当 时，y有最小值且y最小=；如果a&0，那么，当时，y有最大值，且y最大=。告诉最大值或最小值，实际上也是告诉了顶点坐标，同样也可以求出顶点式。例：已知二次函数当x＝4时有最小值－3，且它的图象与x轴两交点间的距离为6，求这个二次函数的解析式。点拨：析解∵二次函数当x＝4时有最小值－3，∴顶点坐标为（4，-3），对称轴为直线x＝4，抛物线开口向上。由于图象与x轴两交点间的距离为6，根据图象的对称性就可以得到图象与x轴两交点的坐标是（1，0）和（7，0）。∴抛物线的顶点为（4，-3）且过点（1，0）。故可设函数解析式为y＝a(x－4)2－3。将（1，0）代入得0＝a(1－4)2－3, 解得a＝13．∴y＝13(x－4)2-3，即y＝13x2－83x＋73。③典型例题三：告诉对称轴，相当于告诉了顶点的横坐标，综合其他条件，也可解出。例如：（1）已知二次函数的图象经过点A（3，-2）和B（1，0），且对称轴是直线x＝3．求这个二次函数的解析式. （2）已知关于x的二次函数图象的对称轴是直线x=1，图象交y轴于点（0，2），且过点（-1，0），求这个二次函数的解析式. （3）已知抛物线的对称轴为直线x=2，且通过点（1，4）和点（5，0），求此抛物线的解析式. （4）二次函数的图象的对称轴x=-4，且过原点，它的顶点到x轴的距离为4，求此函数的解析式．④典型例题四：利用函数的顶点式，解图像的平移等问题非常方便。例：把抛物线y=ax2+bx+c的图像向右平移3 个单位, 再向下平移2 个单位, 所得图像的解析式是y=x2-3x+5, 则函数的解析式为_______。点拨：解先将y=x2-3x+5化为y=(x-32)2+5-94, 即y=(x-32)2+114。∵它是由抛物线的图像向右平移3 个单位, 再向下平移2 个单位得到的，∴原抛物线的解析式是y=(x-32+3)2+114+2=(x+32)2+194=x2+3x+7。直线与圆的位置关系:直线与圆的位置关系有三种：直线与圆相交，直线与圆相切，直线与圆相离。 （1）相交：直线和圆有两个公共点时，叫做直线和圆相交，这时直线叫做圆的割线，公共点叫做交点AB与⊙O相交，d&r； （2）相切：直线和圆有唯一公共点时，叫做直线和圆相切，这时直线叫做圆的切线，这个唯一的公共点叫做切点。AB与⊙O相切，d=r。（3）相离：直线和圆没有公共点时，叫做直线和圆相离,AB与圆O相离，d&r。（d为圆心到直线的距离）直线与圆的三种位置关系的判定与性质： （1）数量法：通过比较圆心O到直线距离d与圆半径的大小关系来判定， 如果⊙O的半径为r，圆心O到直线l的距离为d，则有： 直线l与⊙O相交d&r； 直线l与⊙O相切d=r； 直线l与⊙O相离d&r； （2）公共点法：通过确定直线与圆的公共点个数来判定。 直线l与⊙O相交d&r2个公共点； 直线l与⊙O相切d=r有唯一公共点； 直线l与⊙O相离d&r无公共点 。圆的切线的判定和性质&&& （1）切线的判定定理：经过半径的外端并且垂直于这条半径的直线是圆的切线。 （2）切线的性质定理：圆的切线垂直于经过切点的半径。 切线长：在经过圆外一点的圆的切线上，这点和切点之间的线段的长叫做这点到圆的切线长。 切线长定理：从圆外一点引圆的两条切线，它们的切线长相等，圆心和这一点的连线平分两条切线的夹角。 直线与圆的位置关系判定方法:平面内，直线Ax+By+C=0与圆x2+y2+Dx+Ey+F=0的位置关系判断一般方法是：1.由Ax+By+C=0，可得y=（-C-Ax）/B，（其中B不等于0），代入x2+y2+Dx+Ey+F=0，即成为一个关于x的方程如果b2-4ac&0，则圆与直线有2交点，即圆与直线相交。如果b2-4ac=0，则圆与直线有1交点，即圆与直线相切。如果b2-4ac&0，则圆与直线有0交点，即圆与直线相离。2.如果B=0即直线为Ax+C=0，即x=-C/A，它平行于y轴（或垂直于x轴），将x2+y2+Dx+Ey+F=0化为（x-a）2+（y-b）2=r2。令y=b，求出此时的两个x值x1、x2，并且规定x1&x2，那么：& 当x=-C/A&x1或x=-C/A&x2时，直线与圆相离；当x1&x=-C/A&x2时，直线与圆相交。&相似三角形性质定理：(1)相似三角形的对应角相等。(2)相似三角形的对应边成比例。(3)相似三角形的对应高线的比，对应中线的比和对应角平分线的比都等于相似比。(4)相似三角形的周长比等于相似比。(5)相似三角形的面积比等于相似比的平方。(6)相似三角形内切圆、外接圆直径比和周长比都和相似比相同，内切圆、外接圆面积比是相似比的平方(7)若a/b =b/c，即b2=ac，b叫做a,c的比例中项(8)c/d=a/b 等同于ad=bc.(9)不必是在同一平面内的三角形里①相似三角形对应角相等，对应边成比例.②相似三角形对应高的比，对应中线的比和对应角平分线的比都等于相似比.③相似三角形周长的比等于相似比
定理推论：推论一：顶角或底角相等的两个等腰三角形相似。推论二：腰和底对应成比例的两个等腰三角形相似。推论三：有一个锐角相等的两个直角三角形相似。推论四：直角三角形被斜边上的高分成的两个直角三角形和原三角形都相似。推论五：如果一个三角形的两边和其中一边上的中线与另一个三角形的对应部分成比例，那么这两个三角形相似。推论六：如果一个三角形的两边和第三边上的中线与另一个三角形的对应部分成比例，那么这两个三角形相似。
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