这部分需要会什么？
- 辨认斜边并使用勾股定理
- 熟记两类特殊直角三角形边长比
- 识别向斜边作高得到的相似关系
核心概念与公式
直角三角形两直角边为 \(a,b\)，斜边为 \(c\)，则 \(a^2+b^2=c^2\)。斜边正对直角且最长。\(45^\circ\! -45^\circ\! -90^\circ\) 的边长比为 \(1:1:\sqrt2\)；\(30^\circ\! -60^\circ\! -90^\circ\) 为 \(1:\sqrt3:2\)，其中 1 对应 \(30^\circ\) 的对边。
从直角顶点向斜边作高 \(h\)，将斜边分成 \(p,q\)，则由相似可得 \(h^2=pq\)。每条直角边的平方等于整条斜边乘其邻接的斜边段。使用这些关系前，必须确认高确实垂直于斜边。
做题时怎样使用
- 先标出直角及其对边，再决定是否有特殊角
- 一般情形用勾股，特殊角用边长比，斜边上的高可用相似
- 最后检查斜边最长、长度为正，并保留精确根式直到需要近似
容易混淆的地方
不能把任意两边都当作直角边。特殊三角形的比例按角的对边对应，不能根据图形摆放方向套用。
用一个例题串起方法
知识点教学例题一个直角三角形的斜边为 10，其中一个锐角为 \(30^\circ\)。求两条直角边。
展开解题思路 ＋
\(30^\circ\) 的对边是斜边的一半，为 5；另一条直角边为 \(5\sqrt3\)。检验：\(5^2+(5\sqrt3)^2=100=10^2\)。
本知识点题库
先独立作答，再查看答案与解析。
直角三角形・勾股与高挑战
In quadrilateral KLMN shown, KL = 3, LM = 3, KN = 23, and MN = 23. Diagonals KM and LN (not shown) intersect at point G (not shown), where GK = 1 and GM = 1. If the length of diagonal LN is \( \sqrt{p}+\sqrt{w} \), where p and w are integers, what is the value of \( p+w \)?
参考答案
直角三角形・勾股与高基础
A right triangle has legs with lengths of 28 centimeters and 24 centimeters. What is the length of this triangle's hypotenuse, in centimeters?
参考答案
直角三角形・勾股与高挑战
A right triangle has sides of length \( 2\sqrt{2} \) , \( 6\surd 2 \), and \( \surd 80 \) units. What is the area of the triangle, in square units?
原库参考解析
直角三角形・勾股与高基础
For the right triangle shown, \( a=4 \) and \( b=5 \). Which expression represents the value of c?
原库参考解析
直角三角形・勾股与高挑战
A square is inscribed in a circle. The radius of the circle is \( \frac{20\sqrt{2}}{2} \) inches. What is the side length, in inches, of the square?
原库参考解析
直角三角形・勾股与高挑战
An isosceles right triangle has a hypotenuse of length 58 inches.What is the perimeter, in inches, of this triangle?
原库参考解析
直角三角形・勾股与高基础
For the right triangle shown, a = 4 and b = 5. Which expression represents the value of c?
原库参考解析
直角三角形・勾股与高挑战
A right triangle has legs with lengths of 24 centimeters and 21 centimeters.If the length of this triangle's hypotenuse, in centimeters, can be written in the form
, where d is an integer, what is the value of d?原库参考解析
直角三角形・勾股与高基础
A square is inscribed in a circle. The radius of the circle is
inches. What is the side length, in inches, of the square?原库参考解析
直角三角形・勾股与高挑战
In the figure shown, \( AD=\frac{121}{3} \) and \( AB=\frac{11\sqrt{130}}{3} \) . What is the length of \( DC \) ?
原库参考解析
直角三角形・勾股与高挑战
A right triangle has legs with lengths of 11 centimeters and 9 centimeters. What is the length of this triangle's hypotenuse, in centimeters?
原库参考解析
直角三角形・勾股与高基础
The figure shows the lengths, in inches, of two sides of a right triangle. What is the area of the triangle, in square inches?
原库参考解析
