Questions: 168 Un quadrato e un rettangolo sono equivalenti. Il perimetro del quadrato misura 56 cm e la base del rettangolo è quadrupla dell'altezza. Calcola il perimetro del rettangolo. [70 cm]

Solution Steps

The perimeter of the square is given by \( P = 4s \), where \( s \) is the side length. Given \( P = 56 \) cm, we can find the side length: \[ s = \frac{P}{4} = \frac{56}{4} = 14 \text{ cm} \]

The area of the square is calculated as: \[ A = s^2 = 14^2 = 196 \text{ cm}^2 \]

Let the height of the rectangle be \( h \). The base is given as \( b = 4h \). The area of the rectangle is equal to the area of the square: \[ A = b \cdot h = 4h \cdot h = 4h^2 \] Setting the areas equal gives: \[ 196 = 4h^2 \] Solving for \( h \): \[ h^2 = \frac{196}{4} = 49 \implies h = \sqrt{49} = 7 \text{ cm} \] Now, we can find the base: \[ b = 4h = 4 \cdot 7 = 28 \text{ cm} \]

The perimeter \( P_r \) of the rectangle is given by: \[ P_r = 2(b + h) = 2(28 + 7) = 2 \cdot 35 = 70 \text{ cm} \]

Final Answer