Question

Find the distance between the two points.A(-2,2) and B(4,5)

Answer 1

We will investgate how to quantify the distance between two points expressed on a cartesian coordinate grid.

A distance between two points on a cartesian coordinate grid employs the use of pythagorean theorem. The theorem is applied for all right angled triangles that relates the hypotenuse length of the right angle triangle with the horizontal distance ( Base ) and vertical distance ( height ).

The pythagorean theorem is written as follows:

`H^2=P^2+B^2`

Each point located on a cartesian coordinate grid has a set of cooridnates relaive to the origin. The line that passes through two given points expresses the hypotenuse length. The difference between x coodinates and y-coordintes of the two points represents base ( B ) and height ( P ) respectively.

We can go ahead and write the cartesian form of the pythagorean theorem where,

`\begin{gathered} B=(x_2-x_1) \\ P=(y_2-y_1) \end{gathered}`

Plugging the above relations in the pythagorean statement:

`H^2=(x_2-x_1)^2+(y_2-y_1)^2_{}`

We are given the two points as follows:

`\begin{gathered} A\colon(x_1,y_1)\to\text{ ( -2 , 2 )} \\ B\colon(x_2,y_2)\to\text{ ( 4 , 5 )} \end{gathered}`

We will plug the respective coordinates in the distance formula developed above:

`\begin{gathered} AB\text{ = }\sqrt[]{(4-(-2))^2+(5-2)^2} \\ \\ AB\text{ = }\sqrt[]{(6)^2+(3)^2} \\ \\ AB\text{ = }\sqrt[]{45^{}} \\ \\ AB\text{ = 6.7082 = 3}\sqrt[]{5} \end{gathered}`

Therefore, the distance between the two given points is:

`3\cdot\sqrt[]{5}\text{ = 6.708 ( 3 decimal places )}`