Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 12: Right Triangles and Quadrilaterals

Lesson 3: Types of Quadrilaterals

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students explore special types of quadrilaterals including rhombuses, parallelograms, and trapezoids, learning how to classify each by their side lengths, angles, and parallel sides. Students practice calculating the area of a rhombus using the formula half the product of its diagonals, and apply the base-times-height formula for parallelograms and the trapezoid area formula involving the sum of its two bases. The lesson builds on prior knowledge of rectangles and squares within the AMC 8 curriculum's Chapter 12 on right triangles and quadrilaterals.

Section 1

Types of Quadrilaterals

Property

  • A quadrilateral is a polygon with four sides.
  • A trapezoid is a quadrilateral with one pair of parallel sides.
  • A parallelogram is a quadrilateral with both pairs of opposing sides parallel.
  • A rhombus is a parallelogram with all sides of the same length.
  • A kite is a quadrilateral with two pairs of adjacent sides of the same lengths.
  • A rectangle is a parallelogram with at least one right angle.
  • A square is a rectangle with all sides of the same length.

Examples

  • A garden plot has four sides, with opposite sides parallel and all corners forming right angles. This shape is a rectangle.
  • A kite has two short sides of equal length next to each other, and two long sides of equal length next to each other. This follows the definition of a kite.
  • A slice of cheese is a quadrilateral with only one pair of parallel sides. This makes it a trapezoid.

Explanation

Think of these shapes like a family tree! A square is a special type of rectangle, which is a special type of parallelogram. Each shape inherits traits from the one above it but adds its own unique rule.

Section 2

Properties of Rhombus Diagonals

Property

In a rhombus, the diagonals are perpendicular bisectors of each other. If the diagonals have lengths d1d_1 and d2d_2, they intersect at right angles and each diagonal cuts the other into two equal parts.

Examples

Section 3

Area of a Rhombus

Property

The area of a rhombus equals half the product of its diagonals:

A=d1×d22A = \frac{d_1 \times d_2}{2}

where d1d_1 and d2d_2 are the lengths of the two diagonals.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Types of Quadrilaterals

Property

  • A quadrilateral is a polygon with four sides.
  • A trapezoid is a quadrilateral with one pair of parallel sides.
  • A parallelogram is a quadrilateral with both pairs of opposing sides parallel.
  • A rhombus is a parallelogram with all sides of the same length.
  • A kite is a quadrilateral with two pairs of adjacent sides of the same lengths.
  • A rectangle is a parallelogram with at least one right angle.
  • A square is a rectangle with all sides of the same length.

Examples

  • A garden plot has four sides, with opposite sides parallel and all corners forming right angles. This shape is a rectangle.
  • A kite has two short sides of equal length next to each other, and two long sides of equal length next to each other. This follows the definition of a kite.
  • A slice of cheese is a quadrilateral with only one pair of parallel sides. This makes it a trapezoid.

Explanation

Think of these shapes like a family tree! A square is a special type of rectangle, which is a special type of parallelogram. Each shape inherits traits from the one above it but adds its own unique rule.

Section 2

Properties of Rhombus Diagonals

Property

In a rhombus, the diagonals are perpendicular bisectors of each other. If the diagonals have lengths d1d_1 and d2d_2, they intersect at right angles and each diagonal cuts the other into two equal parts.

Examples

Section 3

Area of a Rhombus

Property

The area of a rhombus equals half the product of its diagonals:

A=d1×d22A = \frac{d_1 \times d_2}{2}

where d1d_1 and d2d_2 are the lengths of the two diagonals.