求连接点 A(2, – 2) 和 B(– 7, 4) 的线段三等分点的坐标

数学是一门与数字和计算相关的学科。并且，根据计算的类型，数学分为不同的分支，如代数、几何、算术等。

几何 它是处理形状及其属性的数学分支。处理涉及坐标的点、线和平面的几何称为坐标几何。

坐标

平面上任意一点的位置可以表示为(x, y)，这些对被称为坐标，x是平面上一点的水平值。这个值也可以称为 x 坐标或横坐标，y 是平面上点的垂直值。该值可以称为 y 坐标或纵坐标。在坐标几何中，点表示在笛卡尔平面上。

笛卡尔平面

它是由两条垂直线组成的平面，即 x 轴（水平轴）和 y 轴（垂直轴）。笛卡尔平面中点的位置可以使用有序对 (x, y) 来表示。

截面公式

如果线 A 和 B 分别具有坐标 (x ₁ , y ₁ ) 和 (x ₂ , y ₂ )，并且 C 是以 m:n 的比率划分线的点，则点 P 的坐标为：

• 当比率 m:n 为内部时：

1. x = (mx ₂ + nx ₁ ) / (m + n)
2. y = (我的₂ + ny ₁ ) / (m + n)

• 当比率在外部时：当点位于线段之外时

1. x = (mx ₂ – nx ₁ ) / (m + n)
2. y = (我的₂ – ny ₁ ) / (m + n)

求连接点 A(2, – 2) 和 B(– 7, 4) 的线段三等分点的坐标。

解决方案：

Point of trisection means the points that can divide a line into three equal parts. According to the question we have a line segment joining A(2, -2) and B (-7, 4) and we have to find the points that can divide the line AB into three parts having an equal length. A line AB as shown in the diagram is given below,

Suppose C and D to be points of intersection,

So, according to the question, the coordinates of C and D are required.

In coordinate, section formula is used to find a point that divides a line into m:n ratio. According to the question, let C and D be the point of trisection having coordinates (x₃, y₃) and (x₄, y₄) respectively. Therefore,

AC = CD = DB

Then, point C will divide the line AB in ratio 1:2. Apply section formula to get the coordinate of point C.

Here,

m = 1, n= 2

x₁ = 2, y₁ = -2

x₂ = -7, y₂ = 4

Apply section formula:

x₃ = (mx₂ + nx₁) /(m + n)

= ((1)(-7) + (2)(2)) / (1 + 2)

=(-7 + 4) / 3

= -3/3

= -1

y₄ = (my₂ + ny₂) / (m + n)

= ((1)(4) + (2)(-2))/ (1+2)

= (4 – 4) / 3

= 0

Coordinate of C(x₃, y₄) = (-1, 0)

C and D divide AB into three equal parts, then CD = DB. Now using section formula or midpoint formula, calculate coordinate of D.

D divides CB in ratio 1:1.

Now for section formula,

m =1, n = 1

x₁ = -1, y₁ = 0

x₂ = -7, y₂ = 4

Apply section formula:

x₄ = (mx₂ + nx₁) /( m + n)

= ((1)(-7) + (1)(-1)) / (1 + 1)

= ( -7 -1 )/ 2

= -8/2

= -4

y₄ = (my₂ + ny₁) / (m + n)

= ((1)(4) + (1)(0))/ (1 + 2)

= (4) / 2

= 2

Coordinate of D(x₄ , y₄) = ( -4, 2)

Therefore, (-1, 0) and (-4, 2) are the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4). 

示例问题

问题1：求连接点A(0, 0) 和B(0, 3) 的线段的三等分点（即三等分的点）的坐标。

解决方案：

Let C and D be the point.

As AC = CD = DB then C will divide the line segment AB in 1:2 ratio.

Here,

m = 1, n= 2

x₁ = 0, y₁ = 0

x₂ = 0, y₂ = 3

Apply section formula:

x₃ = (mx₁ + nx₁) / (m + n)

= ((1)(0) + (2)(0)) / (1 + 2)

= (0) / 3

= 0

y₄ = (my₁ + ny₂) / (m + n)

= ((1)(3) + (2)(0)) / (1 + 2)

= (3) / 3

= 1

Coordinate of C(x₃ , y₃) = (0, 1)

Now using section formula or midpoint formula, calculate coordinate of D. D divides CB in ratio 1:1.

Now for section formula,

m =1, n = 1

x₁ = 0, y₁ = 1

x₂ = 0, y₂= 3

Apply section formula:

x₄= (mx₂ + nx₁) / (m + n)

= ((1)(0) + (1)(0)) / (1+1)

= ( 0 )/ 2

= 0

y₄= (my₂ + ny₁) / (m + n)

= ((1)(3) + (1)(1))/ (1 + 2)

= ( 4) / 2

= 2

Coordinate of D(x₄ , y₄) = ( 0, 2)

问题 2：求以 1:2 的比例划分连接点 A(2, 4) 和 B(8,10) 的线段的点的坐标。

解决方案：

Let C be the point that divides the line segment joining the points A(2, 4) and B(8,10) in 1:2 ratio. Here,

m =1, n = 2

x₁ = 2, y₁ = 4

x₂ = 8, y₂ = 10

Apply section formula,

x= (mx₂ + nx₁) / (m + n)

= ((1)(8) + (2)(2)) / (1 + 2)

= (12 )/ 3

= 4

y = (my₂ + ny₁) / (m + n)

= ((1)(10) + (2)(4))/ (1 + 2)

= (18) / 3

= 6

Coordinate of point that divides line segment AB in 1:2 ratio is (4, 6).