梯形有哪些不同的特性?
测量是数学的一个分支或几何学科,它处理几何图形和参数,如长度、体积、面积、形状、表面积、大小、区域等。简单地说,当我们谈论长度、面积、体积时几何图形的特定形状或不同参数的测量称为测量。
在测量中,形状以 2 维或 3 维形状存在。
测量公式
- 正方形:面积 = a 2 ,周长 = 4a
- 矩形:面积 = L x B,周长 = 2 ( L + B )
- 圆:面积 = π r 2 ,周长 =2πr
- 等腰三角形:面积 = 1/2 xbxh,周长 = a +b + 2c
- 等边三角形:面积 = ( √3/4 ) xa 2 ,周长 = 3a
- 直角三角形:面积 = 1/2 xbxh,周长 = b + 斜边 + h
- 菱形:面积 = 1/2 xd 1 xd 2 ,周长 = 4 x 边
- 平行四边形:面积 = bxh ,周长 = 2 ( l + b )
- 梯形:面积 = 1/2 h ( a + c) ,周长 = a + b + c + d
什么是梯形?
梯形是具有一对平行相对边的闭合形状的二维四边形。梯形的平行边称为底,梯形的不平行边称为腿。梯形有四个边和四个角。平行四边形也称为具有两条平行边的梯形。
在图1中,AB和CD是梯形的基部,AC和BD是梯形的腿。
梯形的类型
根据边和角度,梯形分为三种类型:
- 等腰梯形
- 不等边梯形
- 右梯形
等腰梯形
腿等长的梯形称为等腰梯形。
如图 2 所示,AC 和 BD 的长度相等。
不等边梯形
所有边和角都不相等的梯形称为斜角梯形。
右梯形
成对成直角且彼此相邻的梯形称为直角梯形。
梯形的性质
- 在梯形中,基部彼此平行。
Example – The sides AB and CD are parallel to each other, shown in figure 5.
- 梯形中相邻的内角之和为 180°。
Example – There are two pairs of co-interior angles. One pair is ∠A and ∠D whereas the other pair is ∠B and ∠C. The sum of each pair of co-interior angles is 180°.
- 梯形的所有内角之和总是 360°。
Example – In figure 5, ∠A+∠D is 180° and ∠B+∠C is 180°. Therefore ∠A+∠D +∠B+∠C = 360°.
- 梯形中两条对角线的长度相等。
Example – In figure 6, Diagonal AC = Diagonal BD.
- 在梯形中,两条对角线相互交叉。
Example – AC and BD are intersecting with each other as shown in figure 6.
- 梯形正好有一对平行的对边。
梯形的面积和周长
在图7中,平行边的长度为a、b的梯形分别以高度h为单位。
我们可以通过计算底的平均值并将其结果乘以高度来找到梯形的面积。
因此,
Area of trapezium = ((AB + DC)/2) × AM =((a +b)/2) × h
where AB and CD are the bases of trapezium and AM is the altitude as shown in figure.
梯形的周长是通过计算其所有边的总和得出的。
因此,
Perimeter of trapezium = AB + BC + CD + AD
where AB, BC, CD and AD are the sides of the trapezium.
示例问题
问题 1. 梯形的形状是什么?
回答:
A trapezium is a two-dimensional 2D closed shape having four straight sides, with one pair of parallel sides.
问题 2. 梯形的平行边长 15 厘米和 11 厘米,不平行边各长 5 厘米。计算梯形的周长?
回答:
It is a Isosceles Trapezium because it is clearly mentioned that non parallel sides of length 5 cm each are equal. According to the isosceles trapezium if two non-parallel sides of the trapezium are of equal length then it is known as isosceles trapezium.
Given, a=15 cm, b=11 cm and c= 5 cm
We get,
Perimeter = a+b+2c
P = 15+11+2(5)
P = 15+11+10
P = 36 cm
问题 3. 求边长为 12cm、14cm、16cm 和 18cm 的梯形的周长。
回答:
As we know that the perimeter of a trapezium is given by calculating the sum of all its sides.
P = Sum of all the sides
P = 12 + 14 + 16 + 18
P = 60 cm
Hence, the perimeter of trapezium is 60 cm.
问题 4. 求梯形的面积,平行边之和为 60 厘米,高为 10 厘米。
回答:
Given, the sum of parallel sides 60cm and the height, h =10cm
We know that,
Area of a trapezium, A = 1/2 × Sum of parallel sides × distance between the parallel sides
By substituting the given values,
A =1/2×60×10
A = 30×10
A = 300 cm2.
Therefore, the area of Trapezium =300cm2.
问题 5. 梯形的 5 个属性是什么?
回答:
The 5 properties of a Trapezium are:
1. In trapezium, bases are parallel to each other.
2. A trapezium has supplementary adjacent angles.
3. Only one pair of opposite sides are parallel.
4. The sum of all the interior angles in a trapezium is always 360°.
5. The line that joins the mid-point of the non-parallel sides is always parallel to the bases.