Find area of kite whose diagonals have lenght 40cm & 70cm
The Correct Answer and explanation is :
The area (A) of a kite can be calculated using the lengths of its diagonals. For a kite with diagonals (d_1) and (d_2), the formula is:
[
A = \frac{1}{2} \times d_1 \times d_2
]
Given that:
- (d_1 = 40 \, \text{cm})
- (d_2 = 70 \, \text{cm})
we can substitute these values into the formula:
[
A = \frac{1}{2} \times 40 \times 70 = \frac{1}{2} \times 2800 = 1400 \, \text{cm}^2
]
So, the area of the kite is (1400 \, \text{cm}^2).
Explanation
In geometry, a kite is a quadrilateral with two distinct pairs of adjacent sides that are equal. This shape has an interesting property where the diagonals are perpendicular to each other and intersect at a right angle. These diagonals form four right triangles within the kite, allowing us to calculate its area without knowing the exact side lengths.
Since the diagonals are perpendicular, they effectively divide the kite into two pairs of congruent triangles. By finding the area of these triangles, we can obtain the area of the kite as a whole. Using the formula ( \frac{1}{2} \times d_1 \times d_2 ), we multiply half the product of the diagonals to get the total area.
In this case, substituting (d_1 = 40 \, \text{cm}) and (d_2 = 70 \, \text{cm}) simplifies the calculation. First, we multiply the diagonal lengths, resulting in (2800), and then divide by 2, giving us (1400 \, \text{cm}^2). This method is efficient because we do not need to calculate individual triangle areas but instead apply the simple formula for any kite with known diagonals.
This formula works because the perpendicular diagonals divide the kite into triangles whose combined area equals half the product of these diagonals, a property unique to shapes with this symmetry. Therefore, understanding the formula and applying it correctly makes solving kite-area problems straightforward.