Math Problem Statement

The diagonals of the parallelogram ABCD intersect at point O. Compare the following vectors: (a) AO and OC, (b) BO and OD. Explain your answer.

Solution

The image contains mathematical problems written in Russian. I'll process the content and provide a solution and explanation. Extracting the text, I see a question concerning a parallelogram ABCDABCD, its diagonals, and vector operations.


Translation of Key Problem Section:

  1. The diagonals of the parallelogram ABCDABCD intersect at point OO.
    • You are asked to compare vectors for: a) AO\vec{AO} and OC\vec{OC},
      b) BO\vec{BO} and OD\vec{OD}.
    • Additionally, explain your answer.

Solution Explanation:

In a parallelogram, the diagonals bisect each other at their intersection point. This means:

  • Point OO divides each diagonal into two equal segments: AO=OC,andBO=OD.\vec{AO} = \vec{OC}, \quad \text{and} \quad \vec{BO} = \vec{OD}.

Thus:

  • AO\vec{AO} and OC\vec{OC} are equal in magnitude and direction:
    AO=OC.\vec{AO} = \vec{OC}.
  • Similarly, BO\vec{BO} and OD\vec{OD} are also equal in magnitude and direction:
    BO=OD.\vec{BO} = \vec{OD}.

This is a fundamental property of parallelograms, as their diagonals always bisect each other.


Would you like further clarification, or do you have additional questions?


Related Questions:

  1. How can the diagonal bisection property of a parallelogram be proven geometrically?
  2. What is the significance of the midpoint theorem in this context?
  3. How does this property extend to other quadrilaterals (e.g., rhombuses, rectangles)?
  4. Can vector addition be used to verify the diagonal properties?
  5. How are these properties used in coordinate geometry?

Tip:

When working with parallelograms, remember that both the opposite sides and the diagonals have key symmetry and equality properties. This can simplify complex problems!

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Math Problem Analysis

Mathematical Concepts

Geometry
Vectors
Parallelogram Properties

Formulas

Vector equality and bisection properties

Theorems

Diagonal bisection property of a parallelogram

Suitable Grade Level

Grades 8-10