Calculating Distance and Midpoint Between Two Points

In coordinate geometry, understanding the distance and midpoint between two points on a plane is fundamental. This article explains how to calculate these quantities using the distance formula and the midpoint formula. We will delve into the step-by-step process through an illustrative example.

Understanding the Distance Formula

The distance between two points A(x_1, y_1) and B(x_2, y_2) in a plane can be calculated using the distance formula:

d √[(x_2 - x_1)2 (y_2 - y_1)2]

Example: Points A(4, 3) and B(7, -5)

Let's say we have two points: A(4, 3) and B(7, -5). Our goal is to find the distance between these points and the coordinates of their midpoint.

Step 1: Calculate the Distance

Calculate the differences:

x_2 - x_1 7 - 4 3

y_2 - y_1 -5 - 3 -8

Plug these values into the distance formula:

d √[32 (-8)2] √[9 64] √73

Note: The distance AB is √73, which is approximately 8.544 (rounded to 3 decimal places).

Step 2: Calculate the Midpoint

The midpoint M of a line segment between two points (x_1, y_1) and (x_2, y_2) is given by:

M left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right)

Plug in the coordinates:

M left( frac{4 7}{2}, frac{3 - 5}{2} right) left( frac{11}{2}, frac{-2}{2} right) (5.5, -1)

Note: The coordinates of the midpoint M are (5.5, -1).

Visual Interpretation

Consider point A at x4 and y3, which is 4 units horizontally and 3 units vertically from the origin on the graph. Similarly, point B is at x7 and y-5, meaning it is 7 units horizontally and 5 units vertically but in the negative direction from the origin.

An right angle triangle can be drawn from point A, vertically to -5 (on the y-axis), then horizontally to 7 (on the x-axis) to reach point B. Point C marks the vertex of this triangle. The lengths are:

( AC 8 ) (vertical distance from y3 to y-5, or 3-(-5)8)

( BC 3 ) (horizontal distance from x4 to x7, or 7-43)

Applying the Pythagorean theorem:

( AC^2 BC^2 AB^2 )

( 8^2 3^2 AB^2 )

( 64 9 AB^2 )

( AB^2 73 )

( AB √73 approx 8.544 )

As for the midpoint:

( frac{4 7}{2} 5.5 )

( frac{3 - 5}{2} -1 )

The coordinates of the midpoint are (5.5, -1).

Conclusion

Through both the distance formula and the visual interpretation using a right angle triangle, we have demonstrated how to calculate the distance and midpoint of two points. The distance between points A and B is √73, approximately 8.544, and the coordinates of the midpoint are (5.5, -1).

FAQ

Q: What is the distance between two points?

A: The distance between two points in a plane can be calculated using the distance formula: d √[(x_2 - x_1)2 (y_2 - y_1)2].

Q: How do you find the midpoint of a line segment?

A: The midpoint of a line segment with endpoints (x_1, y_1) and (x_2, y_2) is given by the formula: M left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right).