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
Rectangles ABCD and EFGH are similar The length of each side of EFGH is 6 times the length of the corresponding side of ABCD. The area of ABCD is 96 square units. What is the area, in square units, of EFGH?
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 6 inches, and angles A and D each have an angle measure or 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・全等与比例线段进阶
Rectangle ABCD is similar to rectangle EFGH. The area of rectangle ABCD is 648 square inches, and the area of rectangle EFGH is 72 square inches. The length of the longest side of rectangle ABCD is 36 inches. What is the length, in inches, of the longest side of rectangle EFGH?
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 5 inches, and angles A and D each have an angle measure of 20°. 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
Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 8 times the length of the corresponding side of ABCD. The area of ABCD is 48 square units. What is the area, in square units, of EFGH?
原库参考解析
相似・全等与比例线段挑战
For triangle PQR and triangle STU, angles P and S each measure 30°, QR = 16 and TU= 12. Which additional piece(s) of information is(are) sufficient to prove that triangle PQR is similar to triangle STU ?
I.The length of \( PQ \) is \( \frac{4}{3} \) the length of \( ST \) .
II. The length of \( PR \) is \( \frac{4}{3} \) the length of \( SU \) .
原库参考解析
相似・全等与比例线段进阶
Triangles \( JKL \)and \( PQR \)are congruent, where \( J \) corresponds to \( P \), and \( K \)corresponds to \( Q \). The measure of angle \( J \) is 55°, the measure of angle \( K \)is 45°, and the measure of angle \( L \) is 80°. What is the measure, in degrees, of angle \( P \)? (Disregard the degree symbol when entering your answer.)
原库参考解析
相似・全等与比例线段进阶
The floor of a library has an area of 4,375 square feet. An architect creates a scale model of the floor of the library, where the length of each side of the model is \( \frac{1}{25} \)times the length of the corresponding side of the floor of the actual library. What is the area, in square feet, of the scale model?
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 34 inches. Each side of square A is multiplied by a scale factor of 2 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 \( FGH \) and triangle \( KLM \), the measures of angles \( G \) and \( L \) are each 60°. The lengths of \( GH \) and \( LM \) are each 52 centimeters, and \( \frac{GH}{GF}=\frac{LM}{LK} \). Which additional piece of information would be necessary to prove that triangle \( FGH \) is congruent to triangle \( KLM \)?
原库参考解析
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 59°, and \( AE=28(BD). \)What is the measure of angle \( CAE \) ?
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 8 and 14 units, respectively. The area of \( \triangle LMR \) is 36 square units. What is the area of \( \triangle PQR \), in square units?
原库参考解析
