GK 2016 Q16

Since the area of the square is 36 cm², the length of a side of the square is 6 cm. Since the shaded area is 27 cm², the area not shaded is (36 − 27) cm² = 9 cm². Let \(a\) cm, \(b\) cm and \(c\) cm be the lengths of the parts of the sides shown on the diagram. The area of a triangle is \(\tfrac{1}{2}\) × base × height.

Therefore \(\tfrac{1}{2} \times a \times 6 + \tfrac{1}{2} \times b \times 6 + \tfrac{1}{2} \times c \times 6 = 9\) and hence \(a + b + c = 3\). Therefore, since \((a + b + c) + (p + q + r + s)\) is the sum of the lengths of two sides of the square and so is equal to 12 cm, the value of \(p + q + r + s\) is 9 cm.