7.6 SSS 相似性

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section SSSS 相似性

  Collapse Expand

  SSSS 相似性

  播放段落

  SSS Similarity Theorem
  ::SSS 相似定理

  播放段落

  By definition, two triangles are similar if all their are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you know only that all sides are proportional, that is enough information to know that the triangles are similar. This is called the Similarity Theorem .
  ::按定义,如果两个三角形都是一致的,而相应的边面是正比的,那么两个三角形是相似的。不必检查所有角度和边来判断两个三角形是否相似。事实上,如果你只知道所有边都是正比的,那么信息就足以知道三角形是相似的。这被称为“相似论 ” 。

  播放段落

  SSS Similarity Theorem: If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.
  ::SSS 相似性定理: 如果两个三角形的所有三对对应边是比例的, 那么两个三角形是相似的 。

  播放段落

  If $\begin{align*}\frac{AB}{YZ} = \frac{BC}{ZX} = \frac{AC}{XY}\end{align*}$ , then $\begin{align*}\triangle ABC \sim \triangle YZX\end{align*}$ .
  ::如果ABYBZX=ACXY,那么ZABC=YZX。

  播放段落

  What if you were given a pair of triangles and the side lengths for all three of their sides? How could you use this information to determine if the two triangles are similar?
  ::如果你们获得一对三角形和三边的侧边长度,你们怎么办?你们如何利用这些信息来判断这两个三角形是否相似呢?

  播放段落

  Examples
  ::实例

  播放段落

  For Examples 1 and 2, use the following diagram :
  ::关于例1和例2,请使用下图:

  

  播放段落

  Example 1
  ::例1

  播放段落

  Is $\begin{align*}\triangle DEF \sim \triangle GHI\end{align*}$ ?
  ::?

  播放段落

  Is $\begin{align*}\frac{15}{30} = \frac{16}{33} = \frac{18}{36}\end{align*}$ ?
  ::1530=1633=1836吗?

  播放段落

  $\begin{align*}\frac{15}{30} = \frac{1}{2}, \frac{16}{33} = \frac{16}{33}\end{align*}$ , and $\begin{align*}\frac{18}{36} = \frac{1}{2}\end{align*}$ . $\begin{align*}\frac{1}{2} \neq \frac{16}{33}, \triangle DEF\end{align*}$ is not similar to $\begin{align*}\triangle GHI\end{align*}$ .
  ::1530=12,1633=1633,1836=12。 121633,ZZDEF和GHI并不相似。

  播放段落

  Example 2
  ::例2

  播放段落

  Is $\begin{align*}\triangle ABC \sim \triangle GHI\end{align*}$ ?
  ::是ABCGHI吗?

  播放段落

  Is $\begin{align*}\frac{20}{30} = \frac{22}{33} = \frac{24}{36}\end{align*}$ ?
  ::2030=2233=2436吗?

  播放段落

  $\begin{align*}\frac{20}{30} = \frac{2}{3}, \frac{22}{33} = \frac{2}{3}\end{align*}$ , and $\begin{align*}\frac{24}{36} = \frac{2}{3}\end{align*}$ . All three ratios reduce to $\begin{align*}\frac{2}{3}\end{align*}$ , $\begin{align*}\triangle ABC \sim \triangle GHI\end{align*}$ .
  ::2030 = 23,2233 = 23,2436 = 23 所有三种比率都降至 23, ABC GHI。

  播放段落

  Example 3
  ::例3

  播放段落

  Determine if the following triangles are similar. If so, explain why and write the similarity statement.
  ::确定以下三角形是否相似。 如果是, 请解释原因并写入相似语句 。

  

  播放段落

  We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with the longest sides and work down to the shortest sides.
  ::我们需要找到三角形对应边的比重, 看看它们是否都一样。 从最长的边开始, 工作到最短的边。

  播放段落

  $\begin{align*}\frac{BC}{FD}=\frac{28}{20}=\frac{7}{5}\end{align*}$
  ::BCFD=2820=75

  播放段落

  $\begin{align*}\frac{BA}{FE}=\frac{21}{15}=\frac{7}{5}\end{align*}$
  ::BAFE=2115=75

  播放段落

  $\begin{align*}\frac{AC}{ED}=\frac{14}{10}=\frac{7}{5}\end{align*}$
  ::ACED=1410=75

  播放段落

  Since all the ratios are the same, $\begin{align*}\triangle ABC \sim \triangle EFD\end{align*}$ by the SSS Similarity Theorem.
  ::由于所有比率都是相同的,所以由SSS相似理论的ABC-EFD。

  播放段落

  Example 4
  ::例4

  播放段落

  Find $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ , such that $\begin{align*}\triangle ABC \sim \triangle DEF\end{align*}$ .
  ::查找 x 和 y , 即 ABC 。

  

  播放段落

  According to the similarity statement, the corresponding sides are: $\begin{align*}\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\end{align*}$ . Substituting in what we know, we have $\begin{align*}\frac{9}{6} = \frac{4x-1}{10} = \frac{18}{y}\end{align*}$ .
  ::根据相似性说明,对应的两面是:ABDE=BCEF=ACDF,替代我们所知的96=4x-110=18y。

  播放段落

  $$\begin{align*}\frac{9}{6} &= \frac{4x-1}{10} && \quad \ \frac{9}{6} = \frac{18}{y}\\ 9(10) &= 6(4x-1) && \quad 9y =18(6)\\ 90 &= 24x-6 && \quad 9y = 108\\ 96 &= 24x && \quad \ y = 12\\ x &= 4\end{align*}$$
  ::96=4x-110 96=18y9(10)=6(4x-1)9y=18(6)90=24x-69y=10896=24xy=12x=4

  播放段落

  Example 5
  ::例5

  播放段落

  Determine if the following triangles are similar. If so, explain why and write the similarity statement.
  ::确定以下三角形是否相似。 如果是, 请解释原因并写入相似语句 。

  

  播放段落

  We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with the longest sides and work down to the shortest sides.
  ::我们需要找到三角形对应边的比重, 看看它们是否都一样。 从最长的边开始, 工作到最短的边。

  播放段落

  $\begin{align*}\frac{AC}{ED}=\frac{21}{35}=\frac{3}{5}\end{align*}$
  ::ACED=2135=35

  播放段落

  $\begin{align*}\frac{BC}{FD}=\frac{15}{25}=\frac{3}{5}\end{align*}$
  ::BCFD=1525=35

  播放段落

  $\begin{align*}\frac{AB}{EF}=\frac{10}{20}=\frac{1}{2}\end{align*}$
  ::ABEF=1020=12

  播放段落

  Since the ratios are not all the same, the triangles are not similar.
  ::由于比率不完全相同,三角形并不相似。

  播放段落

  Review
  ::回顾

  播放段落

  Fill in the blanks.
  ::填满空白。

  播放段落
  1. If all three sides in one triangle are __________________ to the three sides in another, then the two triangles are similar.
     ::如果一个三角形的所有三边 都在另一个三角形的三边, 那么两个三角形是相似的。
  播放段落
  2. Two triangles are similar if the corresponding sides are _____________.
     ::两个三角形相似,如果对应的两边是 。

  播放段落

  Use the following diagram for questions 3-5. The diagram is to scale.
  ::问题3-5使用下图。该图将缩放。

  

  播放段落
  3. Are the two triangles similar? Explain your answer.
     ::两个三角形相似吗?
  播放段落
  4. Are the two triangles congruent? Explain your answer.
     ::两个三角形一致吗?
  播放段落
  5. What is the scale factor for the two triangles?
     ::两个三角的尺度系数是多少?

  播放段落

  Fill in the blanks in the statements below. Use the diagram to the left.
  ::填写以下语句中的空白。用图向左。

  

  播放段落
  6. $\begin{align*}\triangle ABC \sim \triangle\end{align*}$ _____
     ::* ABC * * ABC * * * ABC * * * ABC * * ABC * * * ABC * * * ABC * * * ABC * * ABC * * ABC * * ABC * * ABC * * * ABC * * * ABC * *
  播放段落
  7. $\begin{align*}\frac{AB}{?} = \frac{BC}{?} = \frac{AC}{?}\end{align*}$
     ::AB=BC=AC=AC=ABC=ABC=ABC=AC=ABC=ABC=AC=ABC=ABC=ABC=AC=ABC=ABC=ABC=ABC=ABC=ABC=AC? ABB=ABC=ABC=ABC=ABC? ABA? AB=ABC=ABC=ABC=AC? ABA? AB=ABA? AB=BC=ABC=AC=AC? ABA? AB? AB=BA? AB=BC=BC=ABC=AC=AC? AC? AB=ABA? AB=AB=AB=AB=BC=ABC=AC=AC? AC? AB? AB? AB=AB=BC=BC=ABC=AC=AC=AC? AC? AC? AB? AB? AB=AB? AB=AB=AB? AB=AB=AB=AB=ABAAA? AC? AC=AC? AC? AC? AC? AC? AC? AB? AB=AC? AB=AC? AB=AC? AB=AC? AB? AC? AB? AB? AB? AB=AB=AB=AB=AB=AB=AB=AB=AB=AB=AC? AC? AC? AC? AC? AC? AC? AB? AC? AC? AC? ABA? AC?
  播放段落
  8. If $\begin{align*}\triangle ABC\end{align*}$ had an altitude, $\begin{align*}AG = 10\end{align*}$ , what would be the length of altitude $\begin{align*}\overline{DH}\end{align*}$ ?
     ::如果ABC有一个高度,AG=10, 高度长度是多少?
  播放段落
  9. Find the perimeter of $\begin{align*}\triangle ABC\end{align*}$ and $\begin{align*}\triangle DEF\end{align*}$ . Find the ratio of the perimeters.
     ::找到ZABC和ZDEF的周界 找出周界的比例

  播放段落

  Use the diagram to the right for questions 10-15.
  ::用图向右看问题10-15。

  

  播放段落
  10. $\begin{align*}\triangle ABC \sim \triangle\end{align*}$ _____
      ::* ABC * * ABC * * * ABC * * * ABC * * ABC * * * ABC * * * ABC * * * ABC * * ABC * * ABC * * ABC * * ABC * * * ABC * * * ABC * *
  播放段落
  11. Why are the two triangles similar?
      ::为什么这两个三角形相似?
  播放段落
  12. Find $\begin{align*}ED\end{align*}$ .
      ::寻找ED。
  播放段落
  13. $\begin{align*}\frac{BD}{?} = \frac{?}{BC} = \frac{DE}{?}\end{align*}$
      ::BD? = BC=DE吗?
  播放段落
  14. Is $\begin{align*}\frac{AD}{DB} = \frac{CE}{EB}\end{align*}$ true?
      ::ADDB=CEEB是真的吗?
  播放段落
  15. Is $\begin{align*}\frac{AD}{DB} = \frac{AC}{DE}\end{align*}$ true?
      ::ADDB=ACDE是真的吗?

  播放段落

  Find the value of the missing variable(s) that makes the two triangles similar. Assume the sides of length 18 and length 9 correspond to each other.
  ::查找使两个三角形相似的缺失变量值。 将长度 18 和 9 的两边相对应 。

  16. 

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

  播放段落

  Resources
  ::资源