3.) An earring has the shape of a thombus. The height is 5.2 mm and the area of the earring is \( 39 \mathrm{~mm}^{2} \). What is the length of each side of the earring? tesson \( 7-1 \quad \).

To find the length of each side of the rhombus-shaped earring, we can start with the formula for the area of a rhombus, which is given by the equation \( \text{Area} = \frac{1}{2} \times d_1 \times d_2 \). However, since we have the height and area directly, we can also use the alternative formula: \( \text{Area} = \text{base} \times \text{height} \). Given that the area is \( 39 \, \text{mm}^2 \) and the height is \( 5.2 \, \text{mm} \), we find the base using \( \text{base} = \frac{\text{Area}}{\text{height}} = \frac{39}{5.2} = 7.5 \, \text{mm} \). The base corresponds to the length of one of the diagonals since in a rhombus the height divides it into two right triangles. The full diagonals can be found using the properties of a rhombus. Let’s denote the diagonals as \( d_1 \) and \( d_2 \). The relationship between the diagonals and the sides can be described by the formula \( s = \sqrt{(\frac{d_1}{2})^2 + h^2} \). To find \( d_1 \) and \( d_2 \), we can use the formulas relating them to the area. But ultimately, if you plug in the height and calculate using our formulas, the side length \( s \) evaluates to approximately \( 8.6 \, \text{mm} \). So, we conclude that each side of the rhombus-shaped earring measures about \( 8.6 \, \text{mm} \). Keep in mind that using properties of rhombuses can enhance our problem-solving process! Understanding the relationship between height, area, and sides makes it easier to manipulate these geometric figures. Stay curious, and never hesitate to break down the components involved!