Summary

7.4 Summary

Description

Quick Overview

This section summarizes the main concepts in coordinate geometry including the distance formula and section formula.

Standard

The summary encapsulates key concepts in coordinate geometry, such as how to determine the distance between two points in a plane, the coordinate of a point that divides a segment in a given ratio, and how to find the midpoint between two points.

Detailed

In this section, we delve into essential formulas and concepts of coordinate geometry. We learned that the distance between two points P(x1, y1) and Q(x2, y2) in a Cartesian plane is given by the formula

Distance Formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Additionally, we studied that the distance of a point P(x, y) from the origin (0, 0) can be expressed as

From the Origin:
\[ OP = \sqrt{x^2 + y^2} \]

We also explored the section formula, which finds the coordinates of point P dividing the segment joining A and B in the ratio m: n, given by

Section Formula:
\[ P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

Lastly, the midpoint of a line segment connecting points P and Q is calculated as

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

Understanding these formulas lays the foundation for more complex applications and problem-solving in coordinate geometry.

Key Concepts

  • Distance Formula: A method of calculating the distance between two points in a coordinate plane.

  • Section Formula: Used to find a point that divides a line segment in a given ratio.

  • Midpoint Formula: Calculates the average position of two points to find the midpoint.

Memory Aids

🎵 Rhymes Time

  • When finding lengths, remember this song, Use two points, it won’t take long. Just square each part, add and then root, That’s how you find the distance, it’s cute!

📖 Fascinating Stories

  • Once in the land of Geometry, a wise teacher explained that each point has its place, and to find distances between them, it becomes a magical adventure using formulas!

🧠 Other Memory Gems

  • To recall Distance Formula: D = square root of dx² + dy², where dx and dy are changes in x and y.

🎯 Super Acronyms

For the Section Formula, remember 'MAP' - M(x1, x2), A(m, n) = P(p1, p2) where P resembles the Position.

Examples

  • Example 1: For points A(1, 2) and B(3, 4), the distance is calculated as PQ = √[(3-1)² + (4-2)²] = √[2² + 2²] = √8 = 2√2.

  • Example 2: To find the midpoint M of A(4, 6) and B(8, 10), M = [(4+8)/2, (6+10)/2] = (6, 8).

Glossary of Terms

  • Term: Distance Formula

    Definition:

    A formula used to determine the distance between two points in a coordinate plane.

  • Term: Section Formula

    Definition:

    A formula to find the coordinates of a point that divides a segment in a specific ratio.

  • Term: Midpoint Formula

    Definition:

    A formula to calculate the midpoint of a segment between two points.