这部分需要会什么？
- 根据对应角或对应边判定相似
- 按顶点顺序写出正确的边长比例
- 区分相似比例与全等关系
核心概念与公式
相似三角形对应角相等、对应边成比例。常用判定包括两角相等（AA）、两组对应边成比例且夹角相等（SAS）、三组对应边成比例（SSS）。全等图形形状和大小都相同，对应边比为 1。
若 \(\triangle ABC\sim\triangle DEF\)，顺序表示 \(A\leftrightarrow D\)、\(B\leftrightarrow E\)、\(C\leftrightarrow F\)。长度倍率为 \(k\) 时，周长倍率为 \(k\)，面积倍率为 \(k^2\)。
做题时怎样使用
- 先从角标记或题目给出的顶点顺序找对应关系
- 用同一方向写各组边长比，并选已知量最多的一组
- 求面积时将长度倍率平方；需要全等时进一步检查倍率是否为 1
容易混淆的地方
两条边的比值相同，并不足以单独证明三角形相似。不要按图上“左边、右边”机械配对，图形可能旋转或翻转。
用一个例题串起方法
知识点教学例题两个相似三角形的对应边长为 6 和 9。较小三角形的面积为 20，求较大三角形的面积。
展开解题思路 ＋
从小到大的长度倍率为 \(9/6=3/2\)，面积倍率为 \((3/2)^2=9/4\)。较大面积为 \(20\times9/4=45\)。
本知识点题库
先独立作答，再查看答案与解析。
相似・全等与比例线段进阶
For the triangles shown, triangle ABC is dilated by a scale factor of 3 to obtain triangle XYZ, where d=16. what is the measure. in degrees, of angle X ?
原库参考解析
相似・全等与比例线段进阶
Each side of square A has a length of 13 inches. Each side of square A is multiplied by a scale factor of 3 to create square B. What is the length, in inches, of each side of square B?
原库参考解析
相似・全等与比例线段挑战
In triangle ABC and triangle DEF, sides AB and DE each have a side length of 10 inches, and angles A and D each have an angle measure of 40°. Which of the following additional pieces of information is(are) sufficient to prove whether triangle ABC is congruent to triangle DEF?
I. The measures of angles B and C are equal.
II. The lengths of sides AC and DF are equal
III.The lengths of sides BC and EF are equal.
原库参考解析
相似・全等与比例线段挑战
In the figure, \( LQ \) intersects \( MP \) at point R, and \( LM \) is parallel to \( PQ \) . The lengths of \( MR \) and \( RP \) are 9 and 17 units, respectively. The area of \( \triangle LMR \) is 45 square units. What is the area of \( \triangle PQR \), in square units?
原库参考解析
相似・全等与比例线段挑战
The table gives the areas and perimeters of two similar rectangles, where n is a constant.
What is the value of n?
原库参考解析
相似・全等与比例线段挑战
Quadrilateral ABCD is a square with diagonal \( AC \) , where AC= 13.Which of the following statements is true?
原库参考解析
相似・全等与比例线段挑战
The table gives the areas and perimeters of two similar rectangles, where n is a constant.
What is the value of n?
原库参考解析
相似・全等与比例线段进阶
In triangle ABC, AB = 4,290 millimeters(mm) and BC= 4,290 mm. which statement is sufficient to prove that triangle ABC is equilateral?
原库参考解析
相似・全等与比例线段挑战
The table gives the areas and perimeters of two similar rectangles,where n is a constant.
What is the value of n?
原库参考解析
相似・全等与比例线段挑战
Triangles ABC and DEF are congruent, where A corresponds to D, and B and E are right angles. The measure of angle A is 62°. What is the measure of angle F?
原库参考解析
相似・全等与比例线段挑战
Triangle ABC and DEF are congruent, where A corresponds to D, and B and E are right angles. The measure of angle A is 62 °. What is the measure of angle F ?
原库参考解析
相似・全等与比例线段进阶
\( \triangle \)QPR is similar to \( \triangle \)STR . The lengths represented by \( ST \) , \( QP \) , \( PR \) , and \( QR \) in the figure are 5, 7, 24, and 25, respectively. What is the length of \( SR \) ?
原库参考解析
