Question

A rhombus has a side length equal to one of the diagonals. The other diagonal is 4cm longer. Find the area of the rhombus.

Answer 1

The area of the rhombus is 25.83 cm².

Explanation:

The area of a rhombus is given by:

`A = (d_(1) * d_(2))/(2)`

Where:

d₁: is one diagonal

d₂: is the other diagonal = d₁ + 4 cm

We know that one side length of the rhombus is equal to d₁. We can imagine a right triangle inside the rhombus, with the following dimensions:

h: hypotenuse of the right triangle

a: one side of the right triangle

b: is the other side of the right triangle

From the above we know that:

h = d₁

`a = (d_(2))/(2) = (d_(1) + 4)/(2)`

`b = (d_(1))/(2)`

We can find d₁ with Pitagoras:

`h^(2) = a^(2) + b^(2)`

`d_(1)^(2) = ((d_(1) + 4)/(2))^(2) + ((d_(1))/(2))^(2)`

`d_(1)^(2) = (1)/(4)(d_(1)^(2) + 8d_(1) + 16 + d_(1)^(2))`

By solving the above quadratic equation for d₁ and taking the positive solution we have:

`d_(1) = 5.46 cm`

So, d₂ is:

`d_(2) = d_(1) + 4 = 5.46 cm + 4 cm = 9.46 cm`

Now, we can find the area:

`A = (d_(1) * d_(2))/(2) = (5.46 cm * 9.46 cm)/(2) = 25.83 cm^(2)`

Therefore, the area of the rhombus is 25.83 cm².

I hope it helps you!