Solved Examples: Classifying Polygons Using Midpoint Formula of Diagonals

📚 Quick Study Guide

• 📐 Midpoint Formula: The midpoint $M$ of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.
• 🔍 Diagonals: Line segments connecting non-adjacent vertices of a polygon.
• 📏 Parallelogram: A quadrilateral with diagonals that bisect each other (i.e., have the same midpoint).
• ♦️ Rectangle: A parallelogram with diagonals that are congruent (equal in length).
• 🔶 Rhombus: A parallelogram with diagonals that are perpendicular.
• 🔲 Square: A parallelogram that is both a rectangle and a rhombus (diagonals are congruent and perpendicular).
• 💡 General Quadrilateral: If the diagonals do not bisect each other, it is a general quadrilateral.

Practice Quiz

1. What is the midpoint of a line segment with endpoints (1, 4) and (5, 2)?
   1. (3, 3)
   2. (6, 6)
   3. (2, -2)
   4. (4, 1)
2. The diagonals of a quadrilateral have midpoints at (2, 3). What can you conclude about the quadrilateral?
   1. It is a parallelogram.
   2. It is a rectangle.
   3. It is a rhombus.
   4. It is a square.
3. If the endpoints of one diagonal of a quadrilateral are (0, 0) and (4, 4), and the endpoints of the other diagonal are (0, 4) and (4, 0), what is the midpoint of both diagonals?
   1. (2, 2)
   2. (4, 4)
   3. (0, 0)
   4. (0, 4)
4. A quadrilateral has diagonals with different midpoints. What type of quadrilateral is it?
   1. Parallelogram
   2. Rectangle
   3. Rhombus
   4. General Quadrilateral
5. The diagonals of a quadrilateral have the same midpoint (3, 5). If the length of one diagonal is 10 and the other is 6, what can you conclude?
   1. It's a rectangle.
   2. It's a square.
   3. It's a parallelogram, but not necessarily a rectangle or rhombus.
   4. It's a rhombus.
6. The diagonals of a quadrilateral have the same midpoint. Their slopes are negative reciprocals. What is the shape?
   1. Rectangle
   2. Square
   3. Rhombus
   4. Parallelogram
7. The diagonals of a quadrilateral have the same midpoint, and equal length. What shape is it?
   1. Rectangle
   2. Square
   3. Rhombus
   4. Parallelogram