1. Definition of a Kite
A kite is a quadrilateral in which two pairs of adjacent sides are equal. This means the equal sides are next to each other (not opposite).
If the kite is named as \(ABCD\), then typically:
\(AB = AD \quad \text{and} \quad BC = CD\)
Kites resemble the flying kites used during festivals, which helps in remembering their shape easily.
2. Properties of a Kite
Kites have special diagonal and angle properties that make them different from parallelograms, rectangles, or trapeziums.
2.1. Two Pairs of Equal Adjacent Sides
The equal sides touch each other. This is different from parallelograms, where equal sides are opposite.
2.2. One Pair of Opposite Angles are Equal
The angles between the unequal sides are equal:
\(\angle B = \angle D\)
2.3. Diagonals are Perpendicular
The diagonals of a kite intersect at right angles:
\(AC \perp BD\)
2.4. One Diagonal Bisects the Other
The longer diagonal bisects the shorter diagonal into two equal parts.
2.5. One Diagonal Bisects Opposite Angles
Usually the longer diagonal acts as a line of symmetry and divides the kite into two equal halves.
3. Diagonals of a Kite
The diagonals determine the symmetry and area of the kite. One diagonal is usually longer and acts as the 'main axis'.
3.1. Diagonal Properties
- Diagonals meet at right angles.
- The longer diagonal bisects the other.
- The longer diagonal bisects opposite angles.
3.2. Example
If diagonal \(AC = 20\text{ cm}\) and diagonal \(BD = 12\text{ cm}\), then:
\(\text{Area} = \dfrac{1}{2} \times 20 \times 12 = 120\text{ cm}^2\)
4. Area of a Kite
The area of a kite is calculated using the lengths of its diagonals.
4.1. Area Formula
If diagonals are \(d_1\) and \(d_2\):
\(\text{Area} = \dfrac{1}{2} d_1 d_2\)
4.2. Quick Example
If diagonals are \(15\text{ cm}\) and \(18\text{ cm}\):
\(\text{Area} = \dfrac{1}{2}(15)(18) = 135\text{ cm}^2\)
5. Kite in Everyday Life
The shape of a kite appears in crafts, rangoli patterns, logos, and decorative designs. Recognizing its geometric structure helps in solving questions faster.