Diagonals of a parallelogram

3.3.6 Diagonals of a parallelogram

Description

Quick Overview

This section explores the properties of diagonals in a parallelogram, specifically focusing on their intersection and the concept of bisection.

Standard

In this section, we learn that the diagonals of a parallelogram are not generally equal in length, but an important property is that they bisect each other. This fact is reinforced through practical activities and geometric rationale, providing a strong foundation for understanding the structure of parallelograms.

Detailed

Diagonals of a Parallelogram

The diagonals of a parallelogram, such as quadrilateral ABCD, are significant in understanding its geometric properties. While it is known that the diagonals are generally not equal in length, they have a unique and important characteristic: they bisect each other.

Key Concept: Bisection of Diagonals

  1. Definition: To bisect means to divide into two equal parts. In the case of a parallelogram, when the diagonals (e.g., AC and BD) intersect at a point O, they divide each other into two equal segments (AO = CO and BO = DO).
  2. Activity: By folding a cut-out parallelogram to find the midpoints of the diagonals, students can observe that the midpoints coincide at the intersection point, showcasing the bisection property visually.
  3. Geometric Proof: Using congruent triangles formed by the diagonals, specifically triangles AOB and COD, we apply the ASA (Angle-Side-Angle) congruency criterion to establish AO = CO and BO = DO, confirming that diagonals bisect each other.

Understanding this property of diagonals is essential as it lays the groundwork for further exploration of parallelograms and their relationships with other geometric figures.

Similar Questions

  1. Example : HELP is a parallelogram (Lengths are in meters). Given that OE = 5 and HL is 7 more than PE? Find OH.

Solution: If OE = 5 then OP also is 5 (Why?)
PE = 10,
Therefore, HL = 10 + 7 = 17
Hence
\[ OH = \frac{1}{2} \times 17 = 8.5 \; \text{(meters)} \]

  1. Example: HELP is a parallelogram (Lengths are in feet). If OE = 6 and HL is 4 less than PE, find OH.

Solution: If OE = 6 then OP also is 6 (Why?)
PE = 12,
Therefore, HL = 12 - 4 = 8
Hence
\[ OH = \frac{1}{2} \times 8 = 4 \; \text{(feet)} \]

  1. Example: HELP is a parallelogram (Lengths are in inches). Given that OE = 3 and HL is 6 more than PE, find OH.

Solution: If OE = 3 then OP also is 3 (Why?)
PE = 5,
Therefore, HL = 5 + 6 = 11
Hence
\[ OH = \frac{1}{2} \times 11 = 5.5 \; \text{(inches)} \]

  1. Example: HELP is a parallelogram (Lengths are in kilometers). If OE = 7 and HL is twice PE, find OH.

Solution: If OE = 7 then OP also is 7 (Why?)
PE = 4,
Therefore, HL = 2 \times 4 = 8
Hence
\[ OH = \frac{1}{2} \times 8 = 4 \; \text{(kilometers)} \]

  1. Example: HELP is a parallelogram (Lengths are in centimeters). If OE = 2 and HL is 9 more than PE, find OH.

Solution: If OE = 2 then OP also is 2 (Why?)
PE = 6,
Therefore, HL = 6 + 9 = 15
Hence
\[ OH = \frac{1}{2} \times 15 = 7.5 \; \text{(centimeters)} \]

Key Concepts

  • Diagonals bisect each other: The diagonals of a parallelogram intersect at a point that divides each diagonal into two equal segments.

  • Congruent triangles from diagonals: The diagonals create pairs of congruent triangles that help demonstrate the properties of the parallelogram.

Memory Aids

🎵 Rhymes Time

  • In a parallelogram's dance, the diagonals split like chance!

📖 Fascinating Stories

  • Imagine two friends, Ada and B, crossing paths at a park – their meeting point always divides their journey equally. This is how it works with the diagonals in a parallelogram!

🧠 Other Memory Gems

  • B.D.E. – Bisect Diagonal Exits: Remember that the diagonals bisect at their intersection.

🎯 Super Acronyms

P.E.A.C.E. - Parallelogram's Equal Angle Congruency Exists

  • Reflects key properties of diagonals and angles.

Examples

  • In parallelogram ABCD, if the diagonals AC and BD intersect at point O, then AO = CO and BO = DO.

  • If ABCD is a parallelogram, you can fold it along diagonal AC to see that point O is exactly the midpoint of both AC and BD.

Glossary of Terms

  • Term: Diagonal

    Definition:

    A line segment connecting two non-adjacent vertices of a polygon.

  • Term: Bisection

    Definition:

    The division of something into two equal parts.

  • Term: Congruent Triangles

    Definition:

    Triangles that are equal in size and shape, having corresponding sides and angles that are equal.

  • Term: ASA Criterion

    Definition:

    A condition for triangle congruence stating that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.