直角公式
三角形是由三个线段包围的简单封闭几何图形。它有一个三边多边形、3 个顶点和 3 个角。三角形有两种类型:根据边和根据角度的度量。根据边,三角形有3种类型:等边三角形、等腰三角形和不等边三角形。根据其角度的度量,它有3种类型:锐角三角形、钝角三角形和直角三角形。
直角三角形
具有其中一个角的三角形是直角,即 90° 称为直角三角形或直角三角形。与直角相对的一侧是斜边。直角三角形永远不可能是等边三角形,因为它的一个角总是 90 度。在直角等腰三角形中,其他两个角各为 45 度。
直角公式
直角三角形的公式由毕达哥拉斯公式解释。该公式表明,斜边的平方等于其他两条边的平方和。毕达哥拉斯公式如下:
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
i.e. (H)2 = (B)2 + (P)2
其中 H = 斜边,表示与直角相对的直角三角形的边。
B = Base,表示直角三角形所在的一侧。
P = 垂直,这意味着直线形成直角。
毕达哥拉斯定理推导
Consider a right-angled triangle ΔABC at vertex B.
Place BD perpendicular to side AC.
Consider the ΔABC and ΔADB,
In ΔABC and ΔADB,
∠ABC = ∠ADB = 90°
∠A = ∠A ⇢ common
Using triangular similarity criterion AA,
ΔABC to ΔADB
So AD/AB = AB/AC
AB2 = AC × AD ⇢ (1)
Consider ∆ABC and ∆BDC in the figure
∠C = ∠C ⇢ common
∠CDB = ∠ABC = 90°
For the similarity of triangles using an angle angle reference (AA), we have
ΔBDC to ΔABC
So CD/BC = BC/AC
⇒ BC2 = AC × CD ⇢ (2)
From the similarity of triangles, we draw the following conclusions:
∠ADB = ∠CDB = 90°
So, if you draw perpendicular to the hypotenuse from the right vertex of a right triangle, then both triangles form a triangle. The sides of a right angle are similar to each other, just like a whole triangle.
To prove: AC2 = AB2 + BC2
Add Equation (1) and Equation (2)
AB2 + BC2 = (AC × AD) + (AC × CD)
AB2 + BC2 = AC ( AD + CD ) ⇢ (3)
AD + CD = AC,
So substitute this value into Equation (3).
AB2 + BC2 = AC (AC)
Now,
AB2+ BC2 = AC2
So the Pythagorean theorem is proved.
直角三角形的周长公式
直角三角形的周长是所有边的总和。例如,如果 a、b 和 c 是直角三角形的边,则周长为 (a + b + c)。既然这是一个直角三角形,我们可以说它的周长等于两条边的长度和斜边的长度之和。求三角形周长的公式是,
Perimeter of a right triangle = a + b + c
直角三角形的面积公式
直角三角形的面积给出了三角形占据的散布或空间。它等于底乘以三角形高的乘积的一半。因为是二维量,所以用平方单位表示。求直角三角形的面积只需要两条边就是底边和高。使用直角三角形的定义,直角三角形的面积为,
Area of a right triangle = (1/2 × base × height) squared.
直角三角形的性质
直角三角形有多种性质,
- 它没有任何钝角。
- 最大角度为 90 度。
- 最长的边称为斜边。
- 两边遵循毕达哥拉斯公式。
示例问题
问题1:在直角三角形中,如果垂线为4厘米,底边为5厘米,斜边的值是多少?
解决方案:
Given, Perpendicular = 4 cm and Base = 5 cm.
By using Pythagoras Formula ,
(H)2 = (B)2 + (P)2
Where , H = Hypotenuse , B = Base and P = Perpendicular
(H)2 = (5)2 + (4)2
(H)2 = 25 + 16
(H)2 = 41
H = √41 cm.
问题2:在一个直角三角形中,如果斜边是5厘米,底边是4厘米,垂直的值是多少?
解决方案:
Given, Hypotenuse = 5 cm and Base = 4 cm.
By using Pythagoras Formula ,
(H)2 = (B)2 + (P)2
Where , H = Hypotenuse , B = Base and P = Perpendicular
(5)2 = (4)2 + (P)2
25 = 16 + (P)2
(P)2 = 9
P = 3 cm.
问题3:直角三角形有多少个高度?
回答:
The right triangle has three altitudes. The three altitudes of a triangle intersect at the orthocenter which forms an acute triangle is inside the triangle.
问题 4:在直角三角形中,如果斜边是 30 厘米,垂直是 24 厘米,底边的值是多少?
解决方案:
Given, Hypotenuse = 30 cm and Base = 24 cm.
Using Pythagoras Formula,
(H)2 = (B)2 + (P)2
Where , H = Hypotenuse , B = Base and P = Perpendicular
(30)2 = (B)2 + (24)2
900 = (B)2 + 576
(B)2 = 324
B = 18 cm.
问题5:如何找到直角的形状?
回答:
To find the shape of a right angle, place a ruler or set square in the corner of the angle and see if the sides line up. If the angle does not line up with the sides of the ruler or set square then it is not the right angle.
问题6:直角三角形的周长是多少,如果底边是10 m,垂线是8 m,斜边是13 m?
解决方案:
Given, Base = 10 m
Perpendicular = 8 m
Hypotenuse = 13 m
Using Perimeter formula of Right Angled triangle,
P = 10 m + 8 m + 13 m
P = 31 m.
问题 7:直角三角形的面积是多少,如果底边是 10 m,垂线是 8 m,斜边是 18 m?
解决方案:
Given, Base = 10 m
Perpendicular = 8 m
Hypotenuse = 18 m
Using Area formula of Right Angled triangle,
A = 1/2 × Base × Height
A = 1/2 × 10 m × 8 m
A = 80 m / 2
A = 40 m2.
问题8:直角三角形有几个锐角?
回答:
A triangle never has only one acute angle. If a triangle has 1 acute angle the other angles will be either right angles or obtuse angles which is not possible as the sum of interior angles of a triangle is 180 degrees.
问题 9:为什么斜边上的正弦是相反的?
回答:
The sine is always the measure of the opposite side divided by the measure of the hypotenuse. Because the hypotenuse is always on the longest side, the number on the bottom of the ratio will always be larger than that on the top.