Defining a Square
Defining a Square is a Grade 5 math skill from Illustrative Mathematics Chapter 7 (Shapes on the Coordinate Plane) that establishes a square as a quadrilateral with four right angles (90°) and four congruent sides. A square is both a type of rectangle (four right angles) and a type of rhombus (four equal sides), making it the most specific type of parallelogram in the quadrilateral hierarchy.
Key Concepts
Property A square is a quadrilateral with four right angles and four congruent sides. It is a special type of rectangle where all sides are equal in length.
Examples A quadrilateral with vertices at $(0, 0)$, $(4, 0)$, $(4, 4)$, and $(0, 4)$ is a square. A rectangle with a length of 7 cm and a width of 7 cm is a square. A rhombus with four right angles is a square.
Explanation A square is defined as a quadrilateral that has four equal sides and four right angles ($90^\circ$). Because it has four right angles, every square is also a rectangle. However, not all rectangles are squares, as rectangles only need to have opposite sides be equal. A square is the most specific type of parallelogram, as it is both a rectangle and a rhombus.
Common Questions
What is the definition of a square?
A square is a quadrilateral with four right angles (each measuring 90°) and four equal (congruent) sides. It is a special type of rectangle where all sides are the same length.
How is a square different from a rectangle?
Every square is a rectangle (four right angles), but a rectangle is only a square if all four sides are also equal. A rectangle can have sides of different lengths, but a square must have all four sides equal.
What chapter defines squares in Illustrative Mathematics Grade 5?
Defining a Square is covered in Chapter 7 of Illustrative Mathematics Grade 5, titled Shapes on the Coordinate Plane.
Is every rhombus a square?
No. A rhombus has four equal sides, but its angles are not necessarily 90°. A square is a special rhombus that also has four right angles. So every square is a rhombus, but not every rhombus is a square.
How do you identify a square on a coordinate plane?
Plot the four vertices and verify that all four sides are equal in length and all four angles are right angles (90°). You can confirm equal sides by computing distances with the distance formula and right angles by checking perpendicular slopes.