The area \( A \) of a rhombus is one-half the product of the diagonals \( p \) and \( q \).

Write a formula to express the given relation.

A = 2 pq ​

Answered by konrad509 | 2024-06-10

The area A of a rhombus is given by the formula A = ½ × p × q, where p and q are the lengths of the diagonals. To prove that the diagonals of a rhombus are perpendicular using the concept of the scalar (dot) product of vectors, consider two adjacent sides of the rhombus represented as vectors v and w. Since a rhombus is a parallelogram, these sides are equal in length and opposite sides are parallel. The diagonals split the rhombus into four right-angled triangles, and therefore by comparing the areas of these triangles and the parallelogram, we can infer that the diagonals are perpendicular.
To calculate area using vectors, the formula for the area of a parallelogram given by vectors a and b is Area = |a × b|, while the area of a triangle formed by these vectors is half of that, Area of triangle = ½ × |a × b|.

Answered by EdwardRow | 2024-06-24

The area of a rhombus is determined using the formula A = 2 1 ​ × p × q , where p and q are the lengths of its diagonals. This relationship highlights how the area is directly related to the product of the diagonals. The diagonals are always perpendicular, which leads to the division of the rhombus into four right triangles, explaining the half in the formula.
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Answered by EdwardRow | 2024-10-30