# Kite – Quadrilaterals

A **Kite** is a quadrilateral in which four sides can be grouped into two pairs of equal-length sides that are adjacent to each other and the diagonals intersect each other at right angles. The figure shown below represents a kite:

**Properties of Kite**

- Kite has 2 diagonals that intersect each other at right angles.
- A kite is symmetrical about its main diagonal.
- Angles opposite to the main diagonal are equal.
- The kite can be viewed as a pair of congruent triangles with a common base.
- The area of the kite is 1/2 * d
_{1}* d_{2}

### To Prove: The diagonals of kite intersect at right angles (are perpendicular).

**Proof:**

In the figure given above, in ∆ABD and ∆BCD

AB = BC (property of kite)

AD = CD (property of kite)

BD = BD (common side)

Hence ∆ABD ≅ ∆BCD (SSS congruency)

Now, in ∆ABC and ∆ADC

AB = BC (property of kite)

Hence ∆ABC is an isosceles triangle.

AD = CD (property of kite)

Hence ∆ADC is an isosceles triangle.

∠BAO = ∠BCO

BO = BO (common side)

Thus, ∆ABO ≅ ∆BCO (SAS rule of congruency)

Now we know ∠AOB = ∠BOC

Also, ∠AOB + ∠BOC = 180° (Linear pair)

Hence, ∠AOB = ∠BOC = 90°

Hence diagonals of kite intersect at right angles.

### Examples

**Example 1: Find the area of kite whose diagonals are 40 cm and 35 cm?**

**Solution:**

The area of the kite with diagonals as d

_{1}and d_{2}is given as 1/2 * d_{1}* d_{2}.Area = 1/2 * 40 * 35

Area = 700 cm

^{2}Hence the area is 700 cm

^{2}

**Example 2: Find the unknown angles of the given kite?**

**Given that:**

**∠JKL = 100**

**∠KJL = 40**

**Solution:**

As we know that the main diagonal bisects the kite into two halves.

Hence the angles ∠KJL = ∠KLM

Hence ∠KLM = 100

Also since the sum of all angles of the quadrilateral is 360.

Hence ∠JML = 120

**Example 3: The Area of a kite-shaped field is 450 cm² and the length of one of its diagonal is 50 cm. A man wants to cross the field through the other diagonal. Find the distance the man has to travel.**

**Solution: **

Given,

Area of a kite = 450 cm²

Length of one diagonal = 50 cm

We know that the Area of Kite = 1/2 * d

_{1}* d_{2}450 = 1/2 × 50 × d

_{2}d

_{2}= 18 cmHence the other man has to travel a distance of 18 cm.