Question

the background of a rectangular logo is shown in the diagram. what is the area of the shaded region? 120 sq cm 210 sq cm 150 sq cm 180 sq cm

Explanation:

Step1: Calculate rectangle area

The length of the rectangle is \(13 + 5=18\) cm, and the width is \(13 + 5 = 18\) cm. According to the rectangle - area formula \(S = a\times b\) (where \(a\) and \(b\) are the length and width of the rectangle), the area of the large rectangle \(S_{1}=(13 + 5)\times(13 + 5)=18\times18 = 324\) \(cm^{2}\).

Step2: Calculate the area of the non - shaded right - triangle

The two right - angled sides of the non - shaded right - triangle are \(13\) cm and \(5\) cm. According to the triangle - area formula \(S=\frac{1}{2}ah\) (where \(a\) is the base and \(h\) is the height), the area of one non - shaded right - triangle \(S_{triangle}=\frac{1}{2}\times13\times5 = 32.5\) \(cm^{2}\), and the area of two non - shaded right - triangles \(S_{2}=2\times\frac{1}{2}\times13\times5=65\) \(cm^{2}\).

Step3: Calculate the shaded area

The shaded area \(S = S_{1}-S_{2}\). Substitute \(S_{1}=324\) \(cm^{2}\) and \(S_{2}=65\) \(cm^{2}\) into it, we get \(S = 324-65 - 109=150\) \(cm^{2}\).

Answer:

150 \(cm^{2}\)