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# Kites in Mathematics: A Comprehensive Guide for Students

## Introduction

A kite is a simple yet interesting quadrilateral shape often appearing in various mathematical problems and concepts. This article is designed to give students an in-depth understanding of kites, their properties, and how they can be applied to real-life situations. We will cover grade appropriateness, math domain, common core standards, definition, key concepts, illustrative examples, real-life applications, practice tests, and FAQs related to kites.

Kites are generally introduced to students around 4th to 6th grade as they start learning about different quadrilateral shapes and their properties. However, the complexity of problems involving kites can vary, making them relevant for students in higher grades.

## Math Domain

Kites belong to the domain of Geometry, specifically the subdomain of Quadrilaterals, which deals with studying different types of four-sided polygons.

## Applicable Common Core Standards

The concept of kites aligns with the following Common Core Standards:

4.G.A.2: Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.

5.G.B.3: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

6.G.A.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing them into rectangles or decomposing them into triangles and other shapes.

## Definition

A kite is a type of quadrilateral having two pairs of consecutive, non-overlapping sides that are congruent (equal in length). The vertices where the congruent sides meet are called the non-adjacent or opposite vertices. The figure below represents a kite.

## Key Concepts

Diagonals: A kite’s diagonals are perpendicular to one another, and one diagonal is bisected by the other.

Angles: The angles between the congruent sides of a kite are equal.

Perimeter: The perimeter of a kite is the total or sum of all the lengths of the sides.

Area: The area of a kite is one-half the product of its diagonals and can be calculated using the formula: Area = $\frac{d_1 × d_2}{2}$, where d1 and  d2 are the lengths of the diagonals.

## Properties of Kites

• A kite consists of two pairs of congruent sides that are adjacent.

Side AB is congruent to side AD, and side BC is congruent to side DC.

• Two angles are equal where the two pairs of sides meet.
• Diagonals form a cross at a right angle (90°).
• The longer diagonal cuts the other diagonal equally into two parts.

## Discussion with Illustrative Examples

Example 1

Consider a kite ABCD, with AB = BC and AD = CD. The diagonals AC and BD intersect at point E. Also, the diagonals are perpendicular, so ∠BEC = 90°.

Example 2

In the kite ABCD, the angle between the congruent sides is equal, so ∠ABC = ∠ADC.

Example 3

Find the perimeter of a kite with its pairs of equal sides as two and five units.

Solution

The figure below shows a sample illustration of the kite in this example. The perimeter of a kite is the sum of all the sides of the kite. You can calculate the perimeter by adding the sides of each pair.

Perimeter=2+2+5+5

Perimeter=14 units

Therefore, the perimeter of the kite is 14 units.

Example 4

Find the area of the kite given below.

Solution

Lengths of the diagonals are:

d1=2+2=4

d2=6+2=8

A kite’s area is equal to half of the product of its diagonals. Hence, we have,

Area=½ (diagonal 1)(diagonal 2)

Area=½(4)(8)

Area=½(3)(2)

Area=16 square units

Therefore, the area of the kite is 16 square units.

## Examples with Solutions

Example 1 True or false

The diagonals of a kite are always equal in length.

Solution

False; a kite’s two diagonals are not the same length.

Example 2

Given a kite with diagonals 8 cm and 12 cm, calculate its area.

Solution

Area=½ (diagonal 1)(diagonal 2)

Area=½ (8)(12)

Area=½ (96)

Area=48 cm2

Therefore, the area of the kite is 48 cm2.

Example 3

The lengths of a kite’s three sides are three ft., 5 ft, and 3 ft.

a. Find the length of the fourth side.

b. Find the perimeter of the kite.

Solution

a. A kite has two pairs of adjacent equal sides, then the length of the fourth side is 5 ft.

b. To calculate its perimeter, we have,

Perimeter=3+3+5+5

Perimeter=16 ft.

Hence, the perimeter of the kite is 16 ft.

## Real-life Application with Solution

A park is shaped like a kite with 100 meters and 60 meters diagonals. What is the area of the park?

Solution

The lengths of the diagonals are:

diagonal 1=100 meters

diagonal 2=60 meters

Finding the area, we have,

Area=½ (diagonal 1)(diagonal 2)

Area=½ (100)(60)

Area=½ (6000)

Area=3000 m2

Therefore, the area of the park is 3000 m2.

## Practice Test

A. Tell whether the following objects resemble a kite.

B. Calculate the perimeter and area of the given kite.

### How to tell if a quadrilateral is a kite?

A kite has two pairs of consecutive, non-overlapping sides that are congruent (equal in length). The vertices where the congruent sides meet are called the non-adjacent or opposite vertices. A kite also has perpendicular diagonals, where one bisects the other.

### What is the total of a kite’s internal angles?

A kite’s internal angles add up to 360°.

### How many sides does a kite have?

A kite has a total of four sides. The two pairs of sides have equal lengths.

### How many pairs of equal angles does a kite have?

There is only one pair of equal angles in a kite. Two angles are equal where the two pairs of sides meet.

### How do we calculate the perimeter and area of a kite?

The perimeter of a kite can be calculated by adding all the lengths of the sides.

Let us say we have the sides m, n, m, and n; the perimeter of a kite is given by

Perimeter=m+n+m+n=2m+2n.

The area of the kite can be calculated using the formula: Area= $\frac{d_1 × d_2}{2}$, where d1 and d2 are the lengths of the diagonals.