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阅读 在平面直角坐标系中,一次函数$y=-\frac{\sqrt{3}}{3}x + 2$的图像分别与$x$轴、$y$轴相交于点$A$、$B$.若以$AB$为一边的等腰三角形$ABC$的底角为$30^{\circ}$,求点$C$的坐标.
求解本题,首先应思考以$AB$为一边的等腰三角形$ABC$的所有可能情况.等腰三角形$ABC$以$AB$为一边有两种可能:
(1)以$AB$为底边;(2)以$AB$为一腰.然后分别讨论这两类情况,从而不重复、不遗漏地求解点$C$的坐标.
解:由题意,可知点$A$、$B$的坐标分别为$(2\sqrt{3},0)$、$(0,2)$,$\tan\angle BAO=\frac{\sqrt{3}}{3}$,
$\therefore \angle BAO = 30^{\circ}$.
(1)$AB$为等腰三角形$ABC$的底边(图①、②).
如图①,作$CD\perp x$轴,则$CD// BO$.
$\because \angle CBA = \angle BAO = 30^{\circ}$,
$\therefore BC// OA$.
$\therefore CD = BO = 2$.
$\because \angle CAD = 60^{\circ}$,
$\therefore DA = \frac{2}{3}\sqrt{3}$.
$\therefore OD = OA - DA = \frac{4}{3}\sqrt{3}$.
$\therefore$点$C$的坐标为$(\frac{4}{3}\sqrt{3},2)$.
如图②，$\because \angle BCO = \angle CBA + \angle BAC = 60^{\circ}$,$OB = 2$,
$\therefore OC = \frac{2}{3}\sqrt{3}$.
$\therefore$点$C$的坐标为$(\frac{2}{3}\sqrt{3},0)$.
(2)$AB$为等腰三角形$ABC$的一腰(图③、④、⑤、⑥).
如图③,作$CD\perp x$轴,则$CD// BO$.
$\because \angle CBA = \angle BAO = 30^{\circ}$,
$\therefore BC// OA$.
$\therefore CD = BO = 2$,$\angle CAD = 30^{\circ}$.
$\therefore AD = 2\sqrt{3}$.
$\therefore OD = OA + AD = 4\sqrt{3}$.
$\therefore$点$C$的坐标为$(4\sqrt{3},2)$.
如图④，$\because OB = 2$,$OA = 2\sqrt{3}$,$BO\perp AO$,
$\therefore AB = 4$.
$\because BC = AB$,
$\therefore OC = OB + BC = 6$.
$\therefore$点$C$的坐标为$(0,6)$.
如图⑤，$\because BC = BA = 4$,$BO\perp AC$,
$\therefore OC = OA = 2\sqrt{3}$.
$\therefore$点$C$的坐标为$(-2\sqrt{3},0)$.
如图⑥，$\because \angle BAO = 30^{\circ}$,$\angle BAC = 120^{\circ}$,
$\therefore \angle OAC = 90^{\circ}$.
$\because AC = AB = 4$,
$\therefore$点$C$的坐标为$(2\sqrt{3},-4)$.
1. 如图,在$\triangle ABC$中,$\angle ACB = 90^{\circ}$,$AC = BC$,直线$MN$经过点$C$,且$AD\perp MN$,$BE\perp MN$,垂足分别为$D$、$E$.若直线$MN$ 绕点$C$旋转,$DE$、$AD$、$BE$之间有没有等量关系？如具有等量关系,请你写出等量关系,并加以证明.
2. 如图,在四边形$ABCD$中,$\angle A = \angle B = 90^{\circ}$,$AD = 3$,$AB = 7$,$BC = 2$,点$P$在$AB$上,$\triangle BPC$与$\triangle APD$相似.试确定点$P$的位置.
3. 一个二次函数的图像的对称轴是过点$(4,0)$且与$y$轴平行的直线,它与$x$轴两个交点的横坐标、与$y$轴交点的纵坐标都是整数,且以坐标轴上三个交点为顶点的三角形面积为$3$.试写出这个二次函数的表达式.
4. 如图,已知平面直角坐标系中有点$A(-2,0)$、$B(4,0)$,点$P$在一次函数$y = \frac{1}{2}x + \frac{5}{2}$的图像上,且$\triangle ABP$是直角三角形.求点$P$的坐标.
5. 如图,在平面直角坐标系中,已知点$A(4,0)$、$B(0,3)$,若有一个直角三角形与$Rt\triangle AOB$全等,且它们有一条公共边.请写出这个直角三角形未知顶点的坐标.
求解本题,首先应思考以$AB$为一边的等腰三角形$ABC$的所有可能情况.等腰三角形$ABC$以$AB$为一边有两种可能:
(1)以$AB$为底边;(2)以$AB$为一腰.然后分别讨论这两类情况,从而不重复、不遗漏地求解点$C$的坐标.
解:由题意,可知点$A$、$B$的坐标分别为$(2\sqrt{3},0)$、$(0,2)$,$\tan\angle BAO=\frac{\sqrt{3}}{3}$,
$\therefore \angle BAO = 30^{\circ}$.
(1)$AB$为等腰三角形$ABC$的底边(图①、②).
如图①,作$CD\perp x$轴,则$CD// BO$.
$\because \angle CBA = \angle BAO = 30^{\circ}$,
$\therefore BC// OA$.
$\therefore CD = BO = 2$.
$\because \angle CAD = 60^{\circ}$,
$\therefore DA = \frac{2}{3}\sqrt{3}$.
$\therefore OD = OA - DA = \frac{4}{3}\sqrt{3}$.
$\therefore$点$C$的坐标为$(\frac{4}{3}\sqrt{3},2)$.
如图②，$\because \angle BCO = \angle CBA + \angle BAC = 60^{\circ}$,$OB = 2$,
$\therefore OC = \frac{2}{3}\sqrt{3}$.
$\therefore$点$C$的坐标为$(\frac{2}{3}\sqrt{3},0)$.
(2)$AB$为等腰三角形$ABC$的一腰(图③、④、⑤、⑥).
如图③,作$CD\perp x$轴,则$CD// BO$.
$\because \angle CBA = \angle BAO = 30^{\circ}$,
$\therefore BC// OA$.
$\therefore CD = BO = 2$,$\angle CAD = 30^{\circ}$.
$\therefore AD = 2\sqrt{3}$.
$\therefore OD = OA + AD = 4\sqrt{3}$.
$\therefore$点$C$的坐标为$(4\sqrt{3},2)$.
如图④，$\because OB = 2$,$OA = 2\sqrt{3}$,$BO\perp AO$,
$\therefore AB = 4$.
$\because BC = AB$,
$\therefore OC = OB + BC = 6$.
$\therefore$点$C$的坐标为$(0,6)$.
如图⑤，$\because BC = BA = 4$,$BO\perp AC$,
$\therefore OC = OA = 2\sqrt{3}$.
$\therefore$点$C$的坐标为$(-2\sqrt{3},0)$.
如图⑥，$\because \angle BAO = 30^{\circ}$,$\angle BAC = 120^{\circ}$,
$\therefore \angle OAC = 90^{\circ}$.
$\because AC = AB = 4$,
$\therefore$点$C$的坐标为$(2\sqrt{3},-4)$.
1. 如图,在$\triangle ABC$中,$\angle ACB = 90^{\circ}$,$AC = BC$,直线$MN$经过点$C$,且$AD\perp MN$,$BE\perp MN$,垂足分别为$D$、$E$.若直线$MN$ 绕点$C$旋转,$DE$、$AD$、$BE$之间有没有等量关系？如具有等量关系,请你写出等量关系,并加以证明.
2. 如图,在四边形$ABCD$中,$\angle A = \angle B = 90^{\circ}$,$AD = 3$,$AB = 7$,$BC = 2$,点$P$在$AB$上,$\triangle BPC$与$\triangle APD$相似.试确定点$P$的位置.
3. 一个二次函数的图像的对称轴是过点$(4,0)$且与$y$轴平行的直线,它与$x$轴两个交点的横坐标、与$y$轴交点的纵坐标都是整数,且以坐标轴上三个交点为顶点的三角形面积为$3$.试写出这个二次函数的表达式.
4. 如图,已知平面直角坐标系中有点$A(-2,0)$、$B(4,0)$,点$P$在一次函数$y = \frac{1}{2}x + \frac{5}{2}$的图像上,且$\triangle ABP$是直角三角形.求点$P$的坐标.
5. 如图,在平面直角坐标系中,已知点$A(4,0)$、$B(0,3)$,若有一个直角三角形与$Rt\triangle AOB$全等,且它们有一条公共边.请写出这个直角三角形未知顶点的坐标.
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