Question

an earring has a shape of a rhombus the height is 5.2 mm & the area of the earring is 39mm square. what is the length of each side of the earring? ​

Answer 1

7.95mm

Explanation:

To find the length of each side of the rhombus-shaped earring, you can use the formula for the area of a rhombus, which is:

Area = (d1 * d2) / 2

Where:

"d1" and "d2" are the lengths of the diagonals of the rhombus.

In your case, you know the area is 39 square mm, and the height is 5.2 mm. The height is one of the diagonals (d1), and you can solve for the other diagonal (d2) using the formula.

39 = (5.2 * d2) / 2

Now, solve for d2:

39 * 2 = 5.2 * d2

78 = 5.2 * d2

Now, divide both sides by 5.2 to find the length of d2:

d2 = 78 / 5.2

d2 = 15 mm

So, one diagonal (the height) is 5.2 mm, and the other diagonal is 15 mm.

Now, to find the length of each side of the rhombus, you can use the Pythagorean theorem because a rhombus can be split into four right triangles with half of each diagonal as the hypotenuse and the side length as one of the legs.

Let's call the length of each side "s." Using the Pythagorean theorem:

s^2 = (d1/2)^2 + (d2/2)^2

s^2 = (5.2/2)^2 + (15/2)^2

s^2 = (2.6)^2 + (7.5)^2

s^2 = 6.76 + 56.25

s^2 = 63.01

Now, take the square root of both sides to find s:

s ≈ √63.01

s ≈ 7.95 mm

So, each side of the rhombus-shaped earring is approximately 7.95 mm long.

Answer 2

To ascertain the length of each side of the rhombus-esque earring, one may delve into the realms of geometricity. A formula of eminent significance shall be invoked, that which governs the calculation of the area of a rhombus:

Area = (d1 * d2) / 2

Herein, the esteemed "d1" and "d2" shall denote the lengths of the diagonals, the most pivotal constituents of this quadrilateral paradigm.

In the present circumstance, the realm of inquiry bequeaths unto us the area, a magnitude of 39 square millimeters, and the vertical expanse, a height that doth measure 5.2 millimeters. Our noble quest is to unearth the lengths of the sides, which art simultaneously the dimensions of the diagonals.

Let us commence our mathematical odyssey by discerning one of the diagonals, which we shall christen as "d1." The equation thus unfurls:

Area = (d1 * 5.2) / 2

With precision, we isolate "d1":

39 mm² = (d1 * 5.2 mm) / 2

Through the alchemy of multiplication by 2, we unveil:

78 mm² = d1 * 5.2 mm

A judicious division by 5.2 millimeters dost reveal the hallowed length of a singular diagonal, ergo, the expanse of each earring side:

d1 = 78 mm² / 5.2 mm ≈ 15 mm

Hence, the length of each side of the rhombus-adorned earring dost approximate 15 millimeters.