连接点 (2, 3) 和 (3, 7) 的线段除以 Y 轴的比例是多少？

数学是一门与数字和计算相关的学科。并且，根据计算类型，数学分为不同的分支，如代数、几何、算术等。几何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是平面上一点的垂直值。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

除以 y 轴表示位于 y 轴上的点。

位于 y 轴上的点的 x 坐标始终为零。

截面公式

如果一条线 AB 分别具有点 A 和 B 的坐标 (x ₁ , y ₁ ) 和 (x ₂ , y ₂ )，并且线 AB 上有一个点 C 将该线以 m:n 的比例划分，则C可由下式求得，称为截面公式：

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

连接点 (2, 3) 和 (3, 7) 的线段除以 Y 轴的比例是多少？

解决方案：

In the above problem statement, there is one line segment say AB having coordinates (2,3) and (3,7) of A and B respectively. Let’s consider a point C having coordinate (x, y) which lies on the y-axis, then the x coordinate of point C is 0, the coordinates of C can be written as (0, y).

Find the ratio in which this C point will divide the line segment AB. To find the ratio, use the section formula of coordinate geometry. Let’s assume that point C divides AB into a k:1 ratio.

As known, the x-coordinate of point C, we can use it to obtain the value of k and get the ratio. Here,

m = k, n =1

x₁ = 2, y₁ = 3

x₂ = 3, y₂ = 7

x- coordinate using section formula:

x = (mx₂ + nx₁) / (m + n)

After substituting the values,

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

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

0 = 3k + 2

– 2 = 3k 

-2/3 = k

C divides the point in ratio k:1 = -2/3:1, therefore ratio is 2:3

A negative sign represents that point C divides the line segment externally in ratio 2:3.

示例问题

问题1：连接点A和B的线段坐标（0，-2）和（3，1）分别除以X轴的比例是多少？

解决方案：

Let C be the point on X-axis that divides AB in ratio k: 1. As C lies on the x-axis, the y-coordinate of the C should be 0. Therefore, the coordinate of C : (x, 0). As known as the y-coordinate of point C, we can use it to obtain the value of k and get the ratio. Here, 

m = k, n = 1

x₁ = 0, y₁ = -2

x₂ = 3, y₂ = 1

y- coordinate using section formula:

y = (my₂ + ny₁) / (m + n)

After substituting the values,

y = ((k)(1) + (1)(-2))/ (k +1)

0 = ( k – 2)/ (k + 1)

0 = k – 2

2 = k

2 = k

C divides the point in ratio k:1 = 2:1, therefore ratio is 2:1.

问题2：点(2,2)以什么比例分割线段AB，其中(0, 0)和(5, 5)分别是A和B的坐标。

解决方案：

Let (2, 2) divide AB in ratio k: 1. As known the a and y-coordinate of point, use it to obtain the value of k and get the ratio. Here,

m = k, n = 1

x₁ = 0, y₁ = 0

x₂ = 5, y₂ = 5

y- coordinate using section formula:

y = (my₂ + ny₁) / (m + n)

After substituting the values,

2 = ((k)(5) + (1)(0)) / (k + 1)

2 = ( 5k ) / (k+1)

2(k + 1) = 5k

2k + 2 = 5k

2 = 3k

2/3 = k

Hence, (0, 0 ) divides the point in ratio k:1 = 2/3:1, therefore ratio is 2:3.