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・全等与比例线段挑战
In triangle RST, angle T is a right angle, point L lies on \( RS \), point K lies on \( ST \), and \( LK \) is parallel to \( RT \). If the length of \( RT \) is 63 units, the length of \( LK \) is 21 units, and the area of triangle RST is 252 square units, what is the length of \( KT \), in units?
参考答案
相似・全等与比例线段进阶
In the figure shown, triangle \( CAE \) is similar to triangle \( CBD \). The measure of angle\( CBD \) is 57°, and \( AE=24(BD) \) . What is the measure of angle \( CAE \)?
参考答案
相似・全等与比例线段进阶
In triangle ABC, \( DE \) is parallel to \( AC \), BD = AD, BE = 9, and AC = 17. What is the length of \( BC \)?
参考答案
相似・全等与比例线段进阶
Triangles FGH and JKL are congruent. In triangle FGH, FG = 2GH, FG = 28, and FH = 29. In triangle JKL, JK = 2KL. What is the length of segment JL?
参考答案
相似・全等与比例线段挑战
The floor of a store has an area of 4,400 square feet. An architect creates a scale model of the floor of the store, where the length of each side of the model is \( \frac{1}{20} \) times the length of the corresponding side of the floor of the actual store. What is the area, in square feet, of the scale model?
参考答案
相似・全等与比例线段进阶
Triangles ABC and A'B'C' are shown. Triangle ABC is dilated by a scale factor of 3 to form triangle A'B'C'. If the length of \( AB \) is 15, what is the length of \( A'B' \)?
参考答案
相似・全等与比例线段挑战
In the figure shown, triangle CAE is similar to triangle CBD. The measure of angle CBD is 58°, and AE = 23(BD). What is the measure of angle CAE?
参考答案
相似・全等与比例线段进阶
For the triangles shown, triangle ABC is dilated by a scale factor of 5 to obtain triangle XYZ, where \( d=17 \). What is the measure, in degrees, of angle X?
参考答案
相似・全等与比例线段进阶
Triangle \( ABC \) is similar to triangle \( XYZ \), where \( A \), \( B \), and \( C \) correspond to \( X \), \( Y \), and \( Z \), respectively. In triangle \( ABC \), the length of \( AB \) is 120 and the length of \( BC \) is 600. In triangle \( XYZ \), the length of \( YZ \) is 80. What is the length of \( XY \) ?
原库参考解析
相似・全等与比例线段进阶
Triangle \( ABC \) is similar to triangle \( XYZ \), where \( A \), \( B \), and \( C \) correspond to \( X \), \( Y \), and \( Z \), respectively. In triangle \( ABC \), the length of \( AB \) is 130 and the length of \( BC \) is 650. In triangle \( XYZ \), the length of \( YZ \) is 70. What is the length of \( XY \)?
原库参考解析
相似・全等与比例线段挑战
Triangle TUV is dilated by a scale factor of \( \frac{1}{2} \)to obtain triangle T'U'V' (not shown), where T corresponds to T' and U corresponds to U' . What is the measure, in degrees, of angle V'?
原库参考解析
相似・全等与比例线段挑战
Triangles ABC and DEF are similar. Each side length of triangle ABC is 4 times the corresponding side length of triangle DEF. The area of triangle ABC is 270 square inches. What is the area, in square inches, of triangle DEF?
原库参考解析
