Show that the area of a kite is half the product of its diagonals. Class 9

Show that the area of a kite is half the product of its diagonals. Class 9

Question 1.

Show that the area of a kite is half the product of its diagonals. Class 9

i. Using algebra, and

ii. Using geometry.

Solution:

Proof:

i. Using algebra:

Let ∠ABCD be kite with diagonals AC = d₁, BD = d₂

Let diagonals intersect at O.

Diagonals of kite are perpendicular and one bisects the other.

∵ AC ⊥ BD

∴ AC bisects BD

∴ BO = OD = $\frac{d_2}{2}$,

∴ Let AO = x, OC = y

∴ AC = AO + OC = x + y = d₁ ...(i)

The kite is divided into 2 right-angled triangles i.e.,

∆ABD, ∆BCD

∴ Area of ∆ ABD = $\frac{1}{2}$ × Base × height

= $\frac{1}{2}$ × BD × AO

Area of ∆ABD = $\frac{1}{2}$ × d₂ × x ... (ii)

Now,

Area of ∆BCD = $\frac{1}{2}$ × base × height

= $\frac{1}{2}$ × BD × OC

∴ Area of ∆BCD = $\frac{1}{2}$ × d₂ × y ...(iii)

∴ Area of kite = Area of ∆ABD + Area of ∆BCD

$=\left(\frac{1}{2} \times x \times d_2\right)+\left(\frac{1}{2} \times y \times d_2\right)$

...[From (ii) and (iii)]

$\begin{aligned} & =\frac{x d_2}{2}+\frac{y d_2}{2} \\ & =\frac{(x+y) d_2}{2}=\frac{d_1 d_2}{2}\end{aligned}$

...[From (i)]

∴ Area of kite = $\frac{1}{2}$d₁ d₂ i.e., is half the product of its diagonals.

ii. Using geometry:

This method involves visualizing the kite inside a bounding rectangle.

Draw a rectangle around the kite with sides parallel to the diagonals.

Let ABCD be a kite with diagonals BD = d₁ and AC = d₂, intersecting at O.

Draw lines parallel to diagonals to form a rectangle of sides d₁ and d₂, enclosing the kite. So,

Area of rectangle = d₁ × d₂

In a kite:

Diagonals are perpendicular

One diagonal bisects the other

So, the kite is divided into 4 right triangles:

∆ABO, ∆BCO, ∆CDO, ∆DAO

Each of these triangles lies inside one of the four smaller rectangles formed by dividing the big rectangle.

Each triangle occupies half the area of the corresponding rectangle.

So, total area of kite = $\frac{1}{2}$ × area of rectangle

= $\frac{1}{2}$ × d₁ × d₂

Area of kite = $\frac{1}{2}$ × d₁× d₂