8.2 The Mid-point Theorem

Description

Quick Overview

The Mid-point Theorem highlights the relationship between the mid-points of the sides of a triangle and its third side.

Standard

This section introduces the Mid-point Theorem, stating that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half its length. The section also explains the converse of the theorem and demonstrates this through examples.

Detailed

The Mid-point Theorem

In this section, we explore the Mid-point Theorem, which states that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and is equal to half its length. This theorem can be visually understood through practical activities. Following this, the converse theorem is also validated, establishing that if a line is drawn through the mid-point of one side of a triangle parallel to another side, it bisects the third side.

The section leads to engaging examples that apply these theorems to geometric situations, demonstrating the significance of understanding midpoint connections within triangular structures.

Key Concepts

  • Mid-point Theorem: The line segment connecting mid-points of two triangle sides is parallel to the third side and half its length.

  • Converse of the Mid-point Theorem: If a line through a mid-point is parallel to a side, it bisects the opposite side.

Memory Aids

🎵 Rhymes Time

  • Mid-point links, triangle thinks, connects the sides and never shrinks.

📖 Fascinating Stories

  • Imagine a triangle throwing a party with midpoints sitting halfway; they share the secrets of length and parallel sway!

🧠 Other Memory Gems

  • E-M-P: 'E' for EF, 'M' for midpoint, 'P' for parallel.

🎯 Super Acronyms

MPT

  • Midpoint Parallel Theorem!

Examples

  • {'example': 'In triangle ABC, D and E are mid-points of sides AB and AC. Show that DE || BC and DE = 1/2 BC.', 'solution': 'By the Mid-point Theorem, as D and E are midpoints, DE = 1/2 BC and DE || BC.'}

  • {'example': 'In triangle PQR, if line segment EF is drawn through the mid-point of side PR parallel to PQ, prove that it bisects QR.', 'solution': 'Using the converse, since EF || PQ, when extended, AF = CF as confirmed by the properties of parallel lines and congruent triangles.'}

Glossary of Terms

  • Term: Midpoint

    Definition:

    The point that divides a line segment into two equal parts.

  • Term: Parallel Lines

    Definition:

    Lines that run in the same direction and never intersect.

  • Term: Congruent Triangles

    Definition:

    Triangles that are identical in shape and size.

  • Term: Bisect

    Definition:

    To divide into two equal parts.

  • Term: Theorem

    Definition:

    A statement that has been proven based on previously established statements.