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・勾股与高基础
In triangle ABC, angle B is a right angle. The length of side AB is \( 10\sqrt{37} \)and the length of side BC is \( 24\sqrt{37} \). What is the length of side AC?
原库参考解析
直角三角形・勾股与高挑战
A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. The diagonal of the rectangle is twice the length of the shortest side of the rectangle. The area of the rectangle is 1,089\( \sqrt{3} \) square units. What is the length, in units, of the diameter of the circle?
原库参考解析
直角三角形・勾股与高基础
One leg of a right triangle has a length of 43.2 millimeters. The hypotenuse of the triangle has a length of 196.8 millimeters. What is the length of the other leg of the triangle, in millimeters?
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 \( 3\sqrt{d} \), where d is an integer, what is the value of d?
原库参考解析
直角三角形・勾股与高挑战
In the figure above, BD = 6 and AD = 8 . What is the length of \( DC \) ?
原库参考解析
直角三角形・勾股与高挑战
In triangle ABC, the measure of angle B is 90° and BD is an altitude of the triangle. The length of AB is 15 and the length of AC is 23 greater than the length of AB. What is the value of \( \frac{BC}{BD} \) ?
原库参考解析
直角三角形・勾股与高进阶
A right triangle has legs with lengths of 28 centimeters and 20 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
In a right triangle, the measure of one of the acute angles is 51°. What is the measure, in degrees, of the other acute angle?
原库参考解析
直角三角形・勾股与高进阶
In the right triangle shown, what is the value of a?
原库参考解析
直角三角形・勾股与高进阶
The lengths of the legs of a right triangle are shown. Which of the following is closest to the length of the triangle's hypotenuse?
原库参考解析
直角三角形・勾股与高挑战
The line segment shown in the xy-plane represents one of the legs of a right triangle. The area of this triangle is \( 36\sqrt{13} \) square units. What is the length, in units, of the other leg of this triangle?
原库参考解析
直角三角形・勾股与高挑战
In triangle ABC, the measure of angle B is 90° and \( BD \) is an altitude of the triangle. The length of \( AB \) is 15 and the length of \( AC \) is 23 greater than the length of \( AB \) . What is the value of \( \frac{BC}{BD} \) ?
原库参考解析
