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
In triangle DEF, the measure of angle D is 40° and the measure of angle E is 95°. In triangle RST, the measure of angle R is 40° and the measure of angle S is 95°. Which of the following additional pieces of information is needed to determine whether triangle DEF is similar to triangle RST ?
原库参考解析
相似・全等与比例线段进阶
Parallelogram \( ABCD \) is similar to parallelogram \( PQRS \). The length of each side of parallelogram \( PQRS \) is 2 times the length of its corresponding side of parallelogram \( ABCD \). The area of parallelogram \( ABCD \) is 30 square centimeters. What is the area, in square centimeters, of parallelogram \( PQRS \)?
原库参考解析
相似・全等与比例线段进阶
The table gives the perimeters of similar triangles \( TUV \) and \( XYZ \), where \( TU \) corresponds to \( XY \). The length of \( TU \) is 16.
What is the length of \( XY \)?
原库参考解析
相似・全等与比例线段进阶
In the figure, \( LQ \) intersects \( MP \) at point \( R \), and \( LM \) is parallel to \( PQ \). The lengths of \( MR, \) \( LR \) , and \( RP \) are 9, 10, and 19, respectively. What is the length of \( LQ \)?
原库参考解析
相似・全等与比例线段挑战
In isosceles triangle \( ABC \), sides \( AB \) and \( AC \) are congruent. Point \( D \) divides side \( BC \) such that the length of \( BD \) is \( \frac{4}{7} \) of the length of \( BC \). Point \( E \) lies on side \( AB \) and point \( F \) lies on side \( AC \) such that when segments \( DE \) and \( DF \) are drawn, angle \( BED \) is congruent to angle \( CFD \). If the length of \( BE \) is 8, what is the length of \( CF \)?
原库参考解析
相似・全等与比例线段挑战
A property owner wants to find the length x, in feet, across a pond as represented in the figure. \( PR \) intersects \( ST \) at point \( Q \) , and \( ∠PTQ \) is congruent to \( ∠RSQ \). The lengths represented by \( PQ \), \( TQ \), \( QS \), and \( RS \) were determined to be 4,200 feet, 8,400 feet, 1,400 feet, and 1,600 feet, respectively. What is the value of x?
原库参考解析
相似・全等与比例线段进阶
In triangle DEF, the measure of angle D is 46° and the measure of angle E is 104°. In triangle RST, the measure of angle R is 46° and the measure of angle S is 104°. Which of the following additional pieces of information is needed to determine whether triangle DEF is similar to triangle RST?
原库参考解析
相似・全等与比例线段进阶
Triangles ABC and DEF are congruent, where A corresponds to D, and B and E are right angles. The measure of angle A is 17°. What is the measure, in degrees, of angle F?
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 3 times the corresponding side length of triangle DEF. The area of triangle ABC is 170 square inches.What is the area, in square inches, of triangle DEF?
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
The floor of a business office has an area of 1,575 square meters. An architect creates a scale model of the floor of the business office, where the length of each side of the model is \( \frac{1}{15} \) times the length of the corresponding side of the actual floor of the business office. What is the area, in square meters, of the scale model?
原库参考解析
相似・全等与比例线段挑战
Triangle ABC is similar to triangle PQR such that A, B, and C correspond to P, Q, and R, respectively, and angles B and Q are right angles. The measure of ∠A is 35∘ and AC = \( \frac{PR}{5} \) . What is the measure of ∠R ?
原库参考解析
