Diagonals

In geometry, a diagonal is a line segment that connects two non-adjacent vertices of a polygon. The term "diagonal" can be applied to polygons with three or more sides, such as triangles, quadrilaterals, pentagons, etc.

Properties of Diagonals:

Diagonals of a triangle:

For a triangle, there are no diagonals since all three vertices are adjacent to each other.

Diagonals of a quadrilateral:

A quadrilateral has two diagonals that connect opposite vertices.
The diagonals of a parallelogram bisect each other, meaning they divide each other into two equal parts.
The diagonals of a rectangle are equal in length and bisect each other at right angles.

Diagonals of a polygon with five or more sides:

The number of diagonals in a polygon can be calculated using the formula: D = n(n-3)/2, where n is the number of sides of the polygon.

Example Problems:

1. Calculate the number of diagonals in a hexagon.

Solution:

Using the formula D = n(n-3)/2, where n is the number of sides of the polygon, we have:

D = 6(6-3)/2 = 6(3)/2 = 9

So, a hexagon has 9 diagonals.

2. Find the length of the diagonals of a rectangle with sides of 6 cm and 8 cm.

Solution:

Since the diagonals of a rectangle are equal in length, we can use the Pythagorean theorem to find the length d of the diagonal:

d² = 6² + 8²

d = √(6² + 8²) = √(36 + 64) = √100 = 10 cm

Therefore, the length of the diagonal of the rectangle is 10 cm.

Study Guide:

When studying diagonals, it's important to remember the following key points:

Understand the definition of a diagonal and its application in various polygons.
Be familiar with the properties of diagonals in different types of polygons, such as triangles, quadrilaterals, and polygons with five or more sides.
Practice using the formula to calculate the number of diagonals in polygons.
Review the relationship between diagonals and other geometric properties, such as bisecting, equal lengths, and right angles in certain polygons.
Solve problems involving the length and properties of diagonals in different polygons.

By understanding these concepts and practicing related problems, you can develop a strong grasp of the topic of diagonals in geometry.

◂Math Worksheets and Study Guides Fifth Grade. Multiplication

Vocabulary/Answer key

Multiplication

The resources above cover the following skills:

Number and Operations (NCTM)

Compute fluently and make reasonable estimates.

Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.

Algebra (NCTM)

Use mathematical models to represent and understand quantitative relationships.

Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.

Connections to the Grade 5 Focal Points (NCTM)

Number and Operations: Building on their work in grade 4, students extend their understanding of place value to numbers through millions and millionths in various contexts. They apply what they know about multiplication of whole numbers to larger numbers. Students also explore contexts that they can describe with negative numbers (e.g., situations of owing money or measuring elevations above and below sea level).