Distance and Midpoint Formula

Distance and Midpoint Formulas

In this lesson, we will explore two important formulas in geometry: the Distance Formula and the Midpoint Formula. These formulas help us calculate the distance between two points and the midpoint of a line segment connecting two points, respectively.

Outline

• Introduction to the Distance Formula
• Introduction to the Midpoint Formula
• Step-by-step examples for both formulas
• Exercises with solutions
• Homework and revision section

1. Distance Formula

The Distance Formula is used to calculate the distance between two points in a coordinate plane. If you have two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the formula is:

The distance between points \( A \) and \( B \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Example 1: Find the distance between two points

Find the distance between points \( A(1, 2) \) and \( B(4, 6) \).

Using the Distance Formula:

\[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

The distance between points \( A \) and \( B \) is 5 units.

2. Midpoint Formula

The Midpoint Formula helps you find the point that is exactly halfway between two points on a coordinate plane. If you have two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the midpoint \( M \) is:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Example 2: Find the midpoint

Find the midpoint between points \( A(1, 2) \) and \( B(4, 6) \).

Using the Midpoint Formula:

\[ M = \left( \frac{1 + 4}{2}, \frac{2 + 6}{2} \right) = \left( \frac{5}{2}, \frac{8}{2} \right) = (2.5, 4) \]

The midpoint is \( (2.5, 4) \).

Exercises

Exercise 1: Find the distance between points \( P(2, 3) \) and \( Q(7, 8) \).

\[ d = \sqrt{(7 - 2)^2 + (8 - 3)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} \approx 7.07 \text{ units} \]

Exercise 2: Find the midpoint between points \( C(-1, -2) \) and \( D(3, 4) \).

\[ M = \left( \frac{-1 + 3}{2}, \frac{-2 + 4}{2} \right) = (1, 1) \]

Homework

Complete the following exercises:

• Find the distance between points \( E(0, 0) \) and \( F(5, 12) \).
• Find the midpoint between points \( G(2, 4) \) and \( H(6, 8) \).

Revision Section

Make sure to review the Distance and Midpoint formulas before attempting the exercises.

• Practice using the formulas with different sets of points.
• Visualize the points on a graph to better understand the concept.

Watch this Video for More Explanation:

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