如何找到两点之间的关系？

几何无疑是数学中最重要的领域之一。它的重要性与算术和代数一样重要，无论是在我们的日常生活中，还是在数学理论和实践中。几何用于数学理论和问题，以确定两种形式之间的距离、它们占用的空间量、它们的大小和它们的位置。

坐标几何

几何学的一个分支，用于研究点的位置，使用它们的坐标。不仅如此，坐标几何还用于研究两点之间的距离，计算一个点如何分割连接两个点的线段，在笛卡尔平面中寻找形状的区域等。

坐标平面

这种由两条线相交形成的平面，一条垂直，一条水平，在数学上称为坐标平面。它是一个二维平面，以垂直线为 y 轴，水平线为 x 轴。平面中两条线的交点称为原点，用 O 表示。匹配网格上的图形用于检测点。坐标平面可用于绘制点、线等。它充当图表，并产生从一个点到另一个点的精确方向。

如何找到两点之间的关系？

解决方案：

In geometry, there are indefinite number of ways to develop a relation between two points, one of them being the application of distance formula.

Distance Formula

In simple terms, distance is the measurement of how far two things/objects/points are separated. Clearly, the distance formula is a mathematical equation for calculating the distance between two objects/points based on their coordinates. It should be noted that the points in question do not necessarily have to be in the same quadrant. The distance formula is used primarily in the coordinate system in mathematics to determine how far apart two points are in a coordinate plane using their coordinates, making it extremely important in the subject of geometry.

$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Example: Let O be the starting point. If OP=OQ, we have a relationship between two points P and Q in a plane. 

Since this is true for (P,Q) as well as (Q,P) it is a symmetric relation.

Similarly, the equivalence of this relation can also be proved. 

示例问题

问题 1. 如果点 (x, y) 与 (3, 6) 和 (-3, 4) 等距，则求 x 和 y 之间的关系。

解决方案：

Given: Point P(x, y) is equidistant from both A(3, 6) and B(-3, 4).

Using distance formula, distance between P(x, y) and A(3, 6) is given by:

D1 = $\sqrt{(3-x)^2+(6-y)^2}$

Using distance formula, distance between P(x, y) and B(-3, 4) is given by:

D2 = $\sqrt{(-3-x)^2+(4-y)^2}$

Since it is given that D1 = D2, we have:

$\sqrt{(3-x)^2+(6-y)^2} = \sqrt{(-3-x)^2+(4-y)^2}$

Square both sides. Then,

x2 + 9 – 6x + y2 + 36 – 12y = x2 + 9 + 6x + y2 + 16 – 8y

⇒ 12x + 4y = 20

⇒ 3x + y = 5

⇒ 3x + y – 5 = 0

问题 2. 如果点 (x, y) 与 (1, 2) 和 (3, 5) 等距，则 X 和 Y 之间的关系是什么？

解决方案：

Given: Point P(x, y) is equidistant from both A(1, 2) and B(3, 5).

Using distance formula, distance between P(x, y) and A(1, 2) is given by:

D1 = $\sqrt{(1-x)^2+(2-y)^2}$

Using distance formula, distance between P(x, y) and B(3, 5) is given by:

D2 = $\sqrt{(3-x)^2+(5-y)^2}$

Since it is given that D1 = D2, we have:

$\sqrt{(1-x)^2+(2-y)^2} = \sqrt{(3-x)^2+(5-y)^2}$

Square both sides. Then,

x2 + 1 – 2x + y2 + 4 – 4y = x2 + 9 – 6x + y2 + 25 – 10y

⇒ 6x – 2x + 10y – 4y = 9 – 1 + 25 – 4

⇒ 4x + 6y = 29

问题 3. 如果点 P(2, – 3) 和 Q(10, y) 之间的距离为 10 个单位，则求 y。

解决方案：

Given: PQ = 10 units

Using distance formula, we have:

$\sqrt{(2-10)^2+(-3-y)^2} = 10^2$

$⇒ \sqrt{(-8)^2+(y+3)^2} = 100$

Square both sides.

⇒ 64 + (y +3)2 = 100

⇒ (y +3)2 = 36

⇒ y = 3 or −9

问题 4. 如果点 (x, y) 与 (3, 6) 和 (-3, 5) 等距，那么 X 和 Y 之间的关系是什么？

解决方案：

Given: Point P(x, y) is equidistant from both A(3, 6) and B(-3, 5).

Using distance formula, distance between P(x, y) and A(3, 6) is given by:

D1 = $\sqrt{(3-x)^2+(6-y)^2}$

Using distance formula, distance between P(x, y) and B(-3, 5) is given by:

D2 = $\sqrt{(-3-x)^2+(5-y)^2}$

Since it is given that D1 = D2, we have:

$\sqrt{(3-x)^2+(6-y)^2} = \sqrt{(-3-x)^2+(5-y)^2}$

Square both sides. Then,

x2 + 9 – 6x + y2 + 36 – 12y = x2 + 9 + 6x + y2 + 25 – 10y

⇒ 6x + 6x + 12y – 10y = 36 – 25

⇒ 12x + 2y = 11

问题 5. 如果点 (x, y) 与 (3, 6) 和 (-3, 2) 等距，那么 X 和 Y 之间的关系是什么？

解决方案：

Given: Point P(x, y) is equidistant from both A(3, 6) and B(-3, 2).

Using distance formula, distance between P(x, y) and A(3, 6) is given by:

D1 = $\sqrt{(3-x)^2+(6-y)^2}$

Using distance formula, distance between P(x, y) and B(-3, 2) is given by:

D2 = $\sqrt{(-3-x)^2+(2-y)^2}$

Since it is given that D1 = D2, we have:

$\sqrt{(3-x)^2+(6-y)^2} = \sqrt{(-3-x)^2+(2-y)^2}$

Square both sides. Then,

x2 + 9 – 6x + y2 + 36 – 12y = x2 + 9 + 6x + y2 + 4 – 4y

⇒ 12x – 8y = -32

⇒ 3x – 2y = -8