La differenza delle diagonali di un rombo è 42 cm e una ei \( \frac{5}{12} \) dell'altra, Calcola l'area. \( \left[1080 \mathrm{~cm}^{2}\right] \)

To solve the problem, we need to find the lengths of the diagonals of the rhombus and then use them to calculate the area. Let's denote the lengths of the diagonals as: - \( d_1 \): the longer diagonal - \( d_2 \): the shorter diagonal From the problem, we have the following conditions: 1. The difference between the diagonals is \( d_1 - d_2 = 42 \) cm. 2. The shorter diagonal is \( \frac{5}{12} \) of the longer diagonal, which can be expressed as \( d_2 = \frac{5}{12} d_1 \). Now, we can substitute the second equation into the first equation to find \( d_1 \). ### Step 1: Substitute and Solve for \( d_1 \) Substituting \( d_2 \) into the first equation: \[ d_1 - \frac{5}{12} d_1 = 42 \] This simplifies to: \[ \frac{7}{12} d_1 = 42 \] Now, we can solve for \( d_1 \): \[ d_1 = 42 \cdot \frac{12}{7} \] ### Step 2: Calculate \( d_1 \) Let's calculate \( d_1 \): \[ d_1 = 42 \cdot \frac{12}{7} = 72 \text{ cm} \] ### Step 3: Calculate \( d_2 \) Now that we have \( d_1 \), we can find \( d_2 \): \[ d_2 = \frac{5}{12} d_1 = \frac{5}{12} \cdot 72 \] ### Step 4: Calculate \( d_2 \) Calculating \( d_2 \): \[ d_2 = \frac{5 \cdot 72}{12} = 30 \text{ cm} \] ### Step 5: Calculate the Area of the Rhombus The area \( A \) of a rhombus can be calculated using the formula: \[ A = \frac{1}{2} d_1 d_2 \] Substituting the values of \( d_1 \) and \( d_2 \): \[ A = \frac{1}{2} \cdot 72 \cdot 30 \] ### Step 6: Calculate the Area Now, let's calculate the area: \[ A = \frac{1}{2} \cdot 72 \cdot 30 = 1080 \text{ cm}^2 \] Thus, the area of the rhombus is \( 1080 \text{ cm}^2 \).