A Kite is a quadrilateral with two distinct pairs of adjacent sides which are equal. A Kite has two pairs of equal sides in which each pair must be distinct disjoint and must be adjacent to each other, sharing a common vertex. This means, the pairs cannot have a side in common. Only one pair of angle which is the angle between the unequal sides is equal to its opposite angle. The diagonal are perpendicular to each other and longer diagonal bisects the shorter one.

In the figure above, ABCD is a kite with AB and AD as one pair and BC, DC as another pair of sides which are equal and share common vertices A and C respectively. AC and BD are the diagonals of the kite. The diagonal AC bisects the diagonal BD. And, $\angle$ABC = $\angle$ADC.

In real life, kites can be seen being flown in the sky as a game with a different shaped tails.
The length of the diagonals of a kite can be found by using Pythagoras theorem.

Let the diagonals AC = $d_1$ and BD = $d_2$. Then $d_2$ bisects $d_1$ and so AO = CO = $\frac{d_1}{2}$.

Using Pythagoras Theorem, OB = $\sqrt{AB^2 - AO^2}$ = $\sqrt{AB^2 - (\frac{d_1}{2})^2)}$.

OD = $\sqrt{(AD^2 - AO^2)}$ = $\sqrt{AD^2 - (\frac{d_1}{2})^2}$.

From the figure, the diagonal $d_2$ = BD = OB + OD = $\sqrt{AB^2 - (\frac{d_1}{2})^2 )}$ + $\sqrt{AD^2 - (\frac{d_1}{2})^2}$.

Area of a Kite: The area of a kite is half the product of its diagonals. The formula is given by:
Area of Kite = $\frac{d_1 d_2}{2}$  where $d_1$ and $d_2$ are the lengths of the diagonals.
Perimeter of a Kite:
The Perimeter of any polygon is the total distance around the outside of a 2-dimensional shape.

A kite has two pairs of equal sides. In the figure above say AB = AD = a and BC = DC = b then the formula can be stated as:
Perimeter of a Kite = 2a + 2b where a and b are the lengths of each side in each pair of equal sides.

## Is a Rhombus a Kite

Yes. Every Rhombus is a kite, as Rhombus satisfies all the properties of kite. A kite with opposite sides parallel becomes a rhombus. Thus every Rhombus is a Kite but not every Kite is a Rhombus.

## Properties of a Kite

• The angles between the unequal sides and opposite to each other are equal.
• The diagonals of a kite are perpendicular to each other.
• The diagonal of a kite which is longer bisects the diagonal which is shorter.
• One diagonal divides the quadrilateral into two congruent triangles and so becomes the line of symmetry.
• One diagonal bisects a pair of opposite angles which are unequal.

## Kite Math Problems

### Solved Examples

Question 1: Find the Perimeter of the kite given below.

Given that AB = 3cm and AD = 7 cm.
Solution:

The length of one side, AB = 3cm. This is equal to the length of BC as the sides AB and BC are equal.
The length of another side, AD = 7 cm. This is equal to the length of CD as the sides AD and CD are equal.
Perimeter of a Kite = 2a + 2b, where a = AB and b = AD are the lengths of each side in each pair of equal sides.
= 2(3) + 2(7).
= 6 + 14.
Perimeter of the given kite = 20 cm.

Question 2: Find the area of the kite whose diagonals are 4 cm and 6 cm long.
Solution:

Given $d_1$ = 3cm and $d_2$ = 6cm.

Area of Kite = $\frac{d_1 d_2}{2}$ where $d_1$ and $d_2$ are the lengths of the diagonals.

Area of the given kite = $\frac{(3 \times 6)}{2}$ = $\frac{18}{2}$ = 9 sq.cm.

Question 3: Find the length of the diagonal of a kite whose area is 176sqcm and the other diagonal is 16cm.
Solution:

Given $d_1$ = 16cm and area of the kite = 176 sq.cm.

Area of Kite = $\frac{d_1 d_2}{2}$ where $d_1$ and $d_2$ are the lengths of the diagonals.

Diagonal $d_2$ = $\frac{(Area \times 2)}{d_1}$ = $\frac{176 \times 2}{16}$ = 11 $\times$ 2 = 22 cm.

Question 4: One side of a kite is 5 cm less than 7 times the length of another. If the perimeter is 86 cm, find the length of each side of the kite.
Solution:

Let the two unequal sides of a kite be a and b. Then we have b = 7a – 5.
Perimeter of a Kite = 2a + 2b, where a and b are the lengths of each side in each pair of equal sides.
The Perimeter of the given kite = 86 cm.
Thus,                                                                  86 = 2a + 2(7a - 5)
86 = 2a + 14a - 10 = 16a - 10.
16a = 96

a = $\frac{96}{16}$ = 6 cm.

b = 7a – 5 = 7(6) – 5 = 42 – 5 – 37 cms.

The length of each side of the kite is 6 cm, 6 cm, 37cm and 37 cm.

Question 5: In the figure below if AD = CD, $\angle$CDB = $\angle$ADB, Prove that ABCD is a kite.

Solution:

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