2.7 相似性的基本理论
章节大纲
-
Measuring the Height of Tall Objects
::测量高高物体高度I need to find the height of a tall tree – how can I measure it?
::我需要找到一棵高树的高度——我如何衡量它?Finding the height of a tall tree is a problem that foresters, also known as forest scientists, routinely face . A forester manages forests, parks, and rangelands by overseeing restorations, conservation, and timber harvesting. Foresters can determine this using many tools, one of which is the Biltmore stick. The Biltmore stick makes use of similar triangles to estimate the height of trees that are too tall to be measured by hand.
::寻找高树的高度是森林学家(又称森林科学家)通常面临的问题。 福斯特(Afester)管理森林、公园和牧场,监管恢复、养护和木材采伐。 林家可以利用许多工具(其中之一是Biltmore棍子 ) 来决定这一点。 比尔特莫尔(Biltmore)棒使用类似的三角形来估计太高而不能用手测量的树木的高度。A Biltmore stick
::比尔特莫尔棍棒In this section, you will learn about the Fundamental Theorem of Similarity and how it can be applied outside of the classroom.
::在本节中,您将了解“相似性基本理论”以及该理论如何在课堂之外应用。
Understanding the Fundamental Theorem of Similarity
::理解相似性的基本理论The Fundamental Theorem of Similarity states that if a line segment splits a triangle and is parallel to one of the sides of the triangle, it will form two similar triangles.
::相似性的基本理论指出,如果一条线段分裂三角形,并与三角形的侧面平行,它就会形成两个相似的三角形。Use the interactive below to visualize this with different types of triangles.
::使用下面的交互效果来用不同的三角形来直观地显示它。INTERACTIVESimilar Triangles-
Move the outside red dots to see how the similar triangles change
::移动外部红色点以查看相似三角形如何变化 -
Move the middle red dot to change the similarity of the triangles
::移动中间红色点以更改三角形的相似性
+Do you want to reset the PLIX?
Connection to Dilation and Construction
::连接到关系和构造Engineers and architects routinely use the Fundamental Theorem of Similarity when designing houses. One example of this which also makes use of dilations is the design of A-frame houses. This style of house design is popular in remote wooded areas and areas with heavy snowfall.
::工程师和建筑师在设计房屋时经常使用“相似性基本理论”,其中一个例子也利用了“框架房屋”的设计,这种房屋设计风格在偏远的林木地区和大雪地区很流行。Fundamental Theorem of Similarity shown as a house.
::相似性的基本理论显示为一栋房子。For a two-story A-frame house, architects would need to consider how the size of the second story compares to the size of the entire house. Since the line dividing the first and second story is parallel to the base of the house, the second story and entire house are similar triangles by the Fundamental Theorem of Similarity . Thus, t he second story can be thought of as a dilation of the house.
::建筑师需要考虑第二个故事的大小如何与整个房子的大小相比。 由于第一个和第二个故事的分界线与房子的底部平行,第二个故事和整个房子是相似的基本理论的相似三角。 因此,第二个故事可以被认为是房子的放大。When you dilate an image, the dilated image will be similar to the pre-image. Use the interactive to explore the relationship between dilation and the Fundamental Theorem of Similarity.
::当放大图像时,放大图像将与预映像相似。使用互动来探索放大和相似性基本理论之间的关系。INTERACTIVETriangle Dilation-
Move the slider to see how the triangle dilates
::移动滑动器以查看三角形如何膨胀
+Do you want to reset the PLIX?
How The Biltmore Stick Works
::Biltmore棍棒如何运作A Biltmore stick has a variety of markings on it to help find the height, the diameter and how many logs can be made from a tree. The first step of a forester uses in finding the height of a tree is to measure the length of his/her arm using the stick. Next, they hold the stick at the point on the stick that is equal to the length of their arm and point the stick straight upward.
::Biltmore 棍子上有许多标记, 以帮助从树上找到高度、 直径和有多少木头。 一个前酯在找到树高时使用的第一个步骤是用棍子测量其手臂的长度。 下一步, 他们将棍子握在与手臂长度相等的棍子上, 并将棍子直指向上 。Biltmore Stick being used to measure a tree.
::Biltmore 棍子被用来测量一棵树。The forester will stand in a spot far from the tree where the bottom of the ruler lines up with the bottom of the tree and the top of the ruler lines up with the top of the tree. Finally, they measure the distance from where they are standing to the tree. The image below shows measurements taken by a forester.
::前列腺将站在离树很远的地方,在那里标尺的底部与树底相连,标尺的顶部与树顶相连。最后,它们测量了他们站在树边的距离。下面的图象显示一个叉子的测量结果。Biltmore Stick making similar triangles
::Biltmore Stick 制造相似三角形From the information, you can create a proportion to find the height of the tree.
::从该信息中,您可以创建一个比例来查找树的高度。
::2828=x247You should be able to immediately see that the height of the tree will be the same as the distance from the tree: 247 feet. The forester held the stick at the same point as the length of his/her arm to ensure that the distance to the tree was also the height of the tree.
::你应该能马上看到树的高度与树的距离相同:247英尺。前列腺将棍子与手臂的长度放在同一点,以确保树与树的距离也与树的高度相同。Find the height of the trees in the interactive below. Note that the height of the tree will not always be equal to the distance from the tree.
::在下面的交互关系中查找树的高度。请注意,树的高度并不总是等于树的距离。INTERACTIVEThe Biltmore Stick-
Use similarity between the small triangle formed by the Biltmore Stick to find the height of the tree.
::Biltmore Stick 所形成的小三角形的相似性, 用来查找树的高度 。 -
Enter the height of the tree in the box.
::在框中输入树的高度。
+Do you want to reset the PLIX?
Sequences of Transformations
::变换序列Use a sequence of rotations , reflections , translations, and dilations to match up two similar shapes. Can you describe the sequence you used?
::使用一个旋转、 反射、 翻译和放大序列来匹配两个相似的形状。 您可以描述您使用的序列吗 ?INTERACTIVEProving Similarity Through TransformationsFind the sequence of transformations on the blue triangle that will produce the similar triangle in black.
::查找蓝色三角形上的转换序列,以生成黑色的类似三角形。-
Press the buttons on the bottom to perform different transformations on the triangle.
::按下底部的按钮来执行三角形上的不同变换。 -
Drag the red triangle, sliders, and/or points to transform the triangle.
::拖曳红色三角形、滑动符和/或点以转换三角形。 -
Press the "Show" button to see the corresponding sides.
::按下“show”按钮查看相应的边。
+Do you want to reset the PLIX?Remember this!
::记住这个!-
According to the Fundamental Theorem of Similarity (FTS), a line that passes through two sides of a triangle and is parallel to the third side forms two similar triangles.
::根据 " 相似性基本理论 " (FTS),这条线穿过三角形两侧,与第三侧平行,形成两个相似的三角形。 -
Dilating a triangle from any of its
vertices
will result in similar triangles as described by the FTS.
::将三角形从任何顶端插入,将导致与FTS描述的相似的三角形。
-
Move the outside red dots to see how the similar triangles change
