Distance and Midpoint Formulas - Algebra I
Description: In this lesson, we will explore two essential formulas in geometry and algebra: the distance and midpoint formulas. The distance formula helps us find the distance between two points on a coordinate plane, while the midpoint formula finds the exact middle point between two points. We will cover the definitions, formulas, and apply them through multiple examples and exercises.
Lesson Outline
- Introduction to Distance Formula
- Introduction to Midpoint Formula
- Examples
- Exercises with Solutions
- Homework
- Revision Section
Distance Formula
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane can be found using the distance formula:
$$ d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}} $$
Midpoint Formula
The midpoint of the line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the midpoint formula:
$$ M = \left( \frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2} \right) $$
Examples
Example 1: Find the distance between the points \(A(2, 3)\) and \(B(6, 7)\).
Solution:
Using the distance formula:
$$ d = \sqrt{{(6 - 2)^2 + (7 - 3)^2}} = \sqrt{{4^2 + 4^2}} = \sqrt{{16 + 16}} = \sqrt{{32}} \approx 5.66 $$
Example 2: Find the midpoint of the segment connecting points \(A(1, 4)\) and \(B(5, 8)\).
Solution:
Using the midpoint formula:
$$ M = \left( \frac{{1 + 5}}{2}, \frac{{4 + 8}}{2} \right) = (3, 6) $$
Exercises
Exercise 1: Find the distance between points \(P(1, 2)\) and \(Q(4, 6)\).
Solution: Distance = 5
Exercise 2: Find the midpoint between \(R(3, 5)\) and \(S(7, 9)\).
Solution: Midpoint = (5, 7)
Homework
Question 1: Calculate the distance between \(X(-3, 2)\) and \(Y(4, -1)\).
Question 2: Determine the midpoint between \(M(2, -2)\) and \(N(6, 4)\).
Revision
To review the concepts of the distance and midpoint formulas, watch the video below for an in-depth explanation:
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