Answer:
2880 square units
Explanation:
Since the area of a rhombus, A = d₁ × d₂/2 where d₁ and d₂ are its diagonals. Given that d₁:d₂ = 2:1, d₁/d₂ = 2/1, so d₁ = 2d₂. Also, the length of side of the rhombus L = 60 units.
We find the length of the diagonal by Pythagoras' theorem since the diagonals are perpendicular to each other and bisect each other at their mid point.
The mid point of d₁ = d₁/2 and the midpoint of d₂ = d₂/2.
By Pythagoras' theorem,
L² = (d₁/2)² + (d₂/2)²
L² = d₁²/4 + d₂²/4
Now, d₁ = 2d₂
So, L² = (2d₂)²/4 + d₂²/4
L² = 4d₂²/4 + d₂²/4
L² = d₂² + d₂²/4
L² = 5d₂²/4
d₂² = 4L²/5
taking square root of both sides,
√d₂² = √(4L²/5)
d₂ = 2L/√5
Since L = 60 units, then
d₂ = 2L/√5
d₂ = 2(60)/√5
d₂ = 120/√5
rationalizing the denominator by multiplying the numerator and denominator by √5, we have
d₂ = 120/√5 × √5/√5
d₂ = 120√5/5
d₂ = 24√5 units
So, our area A = d₁ × d₂/2
Since d₁ = 2d₂, our area is
A = 2d₂ × d₂/2
A = d₂ × d₂
A = d₂²
substituting d₂ = 24√5, we have
A = d₂²
A = (24√5)²
A = (24)²(√5)²
A = 576 × 5
A = 2880 square units