Question

The length of one diagonal of a rhombus is 2 times the length of the other diagonal. Write an expression that represents the

perimeter of the rhombus, where d represents the shorter diagonal.

Answer 1

Explanation:

In a rhombus, all sides are equal in length. Let's denote the length of each side by s. We can use the Pythagorean theorem to relate the diagonals and the sides of the rhombus. If d is the length of the shorter diagonal, then the length of the longer diagonal is 2d. The Pythagorean theorem gives us:

(s/2)^2 + d^2 = s^2

and

(s/2)^2 + (2d)^2 = s^2

Simplifying and solving for s, we get:

s = √(4d^2 - d^2) = √3d

The perimeter of the rhombus is the sum of the four sides, so we have:

perimeter = 4s = 4√3d

Therefore, the expression that represents the perimeter of the rhombus in terms of the length of the shorter diagonal (d) is 4√3d.

Answer 2

The perimeter of a rhombus is given by 4 times the length of one of its sides.

Use the Pythagorean theorem to relate the two diagonals of the rhombus:

The length of the longer diagonal (D) is related to the length of the shorter diagonal (d) as follows:

D^2 = d^2 + d^2 = 2d^2

We also know that the length of the longer diagonal is twice the length of the shorter diagonal:

D = 2d

Substituting the second equation into the first, we get:

(2d)^2 = 2d^2

4d^2 = 2d^2

2d^2 = 4d^2/2

d^2 = 2d^2/4

d^2 = d^2/2

Multiplying both sides by 2:

2d^2 = d^2

d^2 = 2d^2

Taking the square root of both sides:

d = sqrt(2)d

So the length of the shorter diagonal is d and the length of the longer diagonal is 2d.

The perimeter of the rhombus is:

P = 4s

where s is the length of one of the sides of the rhombus. We can express s in terms of d using the Pythagorean theorem:

s^2 = (d/2)^2 + d^2

s^2 = d^2/4 + d^2

s^2 = 5d^2/4

s = sqrt(5)d/2

Substituting this expression for s into the formula for the perimeter, we get:

P = 4s = 4(sqrt(5)d/2) = 2sqrt(5)d