# Properties of a Kite: Sides, Angles & Diagonals

#Geometry

## TL;DR
A kite is a quadrilateral with two pairs of adjacent equal sides, diagonals that cross at right angles, and one pair of equal opposite angles. This article covers every property of a kite by sides, angles, diagonals, and symmetry, derives the area formula \( \frac{1}{2} \times d_1 \times d_2 \), and works through six examples.

## What Is a Kite in Geometry?
A **kite** is a quadrilateral (a four-sided shape) with **two pairs of adjacent sides that are equal in length**. _Adjacent_ means the equal sides sit next to each other and share a vertex, not opposite each other across the shape.

In kite ABCD, the two short sides AB and AC are equal, and the two long sides BD and CD are equal. The shape has one **axis of symmetry**, the long diagonal AD: fold along it and the two halves match exactly. Every property below is a consequence of that one mirror line.

## Properties of a Kite: Sides
- **Two pairs of adjacent equal sides.** AB = AC and BD = CD.
- **No parallel sides.** Unlike a parallelogram or trapezium, a kite has no pair of parallel sides.
- **Four sides in total**, with two distinct side lengths in general.

A square is a special case where all four sides are equal, so technically every square satisfies the kite condition.

## Properties of a Kite: Angles
- **One pair of equal opposite angles.** The angles between the unequal sides are equal: \( \angle B = \angle C \).
- **The other pair is generally unequal.** \( \angle A \) and \( \angle D \) are usually different from each other.
- **The interior angles sum to 360°**, as in every quadrilateral: \( \angle A + \angle B + \angle C + \angle D = 360° \).

## Properties of a Kite: Diagonals
- **The diagonals cross at right angles.** Diagonals AB and CD meet at 90°.
- **The longer diagonal bisects the shorter one.** The axis of symmetry AD cuts BC into two equal halves at the crossing point O.
- **The shorter diagonal splits the kite into two isosceles triangles**, while the longer diagonal splits it into two congruent triangles.

## How Do You Find the Area of a Kite?
The area depends only on the two diagonals:

\( A = \frac{1}{2} \times d_1 \times d_2 \)

where \( d_1 \) and \( d_2 \) are the lengths of the two diagonals. The kite fills exactly half of the rectangle formed by the diagonals.

| Symbol | Meaning | Units |
| --- | --- | --- |
| A | Area enclosed by the kite | square units (cm²) |
| d1 | Length of the longer diagonal | length units (cm) |
| d2 | Length of the shorter diagonal | length units (cm) |

The **perimeter** is simpler: if \( a \) is the short side and \( b \) is the long side, then \( P = 2a + 2b = 2(a + b) \).

## Is a Kite a Parallelogram?
**No, a kite is not a parallelogram.** A parallelogram needs two pairs of _parallel_ sides; a kite has _no_ parallel sides.

## Examples of the Properties of a Kite

### Example 1
In kite ABCD, AB = 6 cm and the side adjacent to it is AC. Find AC.

By the adjacent-equal-sides property, \( AC = AB = 6 \text{ cm} \).

### Example 2
A kite has diagonals of length 12 cm and 5 cm. Find its area.

Correctly: \( A = \frac{1}{2} \times 12 \times 5 = 30 \text{ cm}² \).

### Example 3
In a kite, three of the four interior angles are \( \angle A = 100° \), \( \angle B = 70° \), and \( \angle C = 70° \). Find \( \angle D \).

The interior angles sum to 360°. Thus, \( \angle D = 360° - (100° + 70° + 70°) = 120° \).

### Example 4
A kite has a short side pair of 5 cm and a long side pair of 8 cm. Find its perimeter.

\( P = 2(5+8) = 26 \text{ cm} \).

### Example 5
A kite has area 54 cm² and one diagonal of length 12 cm. Find the other diagonal.

Work backward: 54 = 6d2, so \( d_2 = 9 \text{ cm} \).

### Example 6
The diagonals of a kite cross at point O. The shorter diagonal is 10 cm long. Find the length of each half of the shorter diagonal.

Because it is bisected, each half is 5 cm.

## Conclusion
- The defining **property of a kite** is two pairs of adjacent equal sides, which forces every other property.
- Its diagonals meet at right angles, and the longer diagonal bisects the shorter one and the kite's two opposite angles.
- The area is \( \frac{1}{2} \times d_1 \times d_2 \), half the product of the diagonals.
- A kite is _not_ a parallelogram: its equal sides are adjacent, and it has no parallel sides.

## Frequently Asked Questions
What are the main properties of a kite?
Two pairs of adjacent equal sides, diagonals that cross at right angles, the longer diagonal bisecting the shorter, one pair of equal opposite angles, and one axis of symmetry along the longer diagonal.

How many lines of symmetry does a kite have?
One. It runs along the longer diagonal.

Are the diagonals of a kite equal?
No. They are generally different lengths.

What is the area of a kite?
\( A = \frac{1}{2} \times d_1 \times d_2 \), half the product of the two diagonals.

Is a square a kite?
Yes, in the family sense. A square satisfies the kite condition.