Here we will get the ideas how to solve the problems on finding the perimeter and area of irregular figures.
1. The figure PQRSTU is a hexagon.
PS is a diagonal and QY, RO, TX and UZ are the respective distances of the points Q, R, T and U from PS. If PS = 600 cm, QY = 140 cm, RO = 120 cm, TX = 100 cm, UZ = 160 cm, PZ = 200 cm, PY = 250 cm, PX = 360 cm and PO = 400 cm. Find the area of the hexagon PQRSTU.
Solution:
Area of the hexagon PQRSTU = area of ∆PZU + area of
trapezium TUZX + area of ∆TXS + area of ∆PYQ + area of trapezium QROY + area of
∆ROS
= {\(\frac{1}{2}\) × 200 × 160 + \(\frac{1}{2}\) (100 + 160)(360 – 200) + \(\frac{1}{2}\) (600 – 360) × 100 + \(\frac{1}{2}\) × 250 × 140 + \(\frac{1}{2}\) (120 + 140) (400 – 250) + \(\frac{1}{2}\) (600 – 400) × 120} cm\(^{2}\)
= (16000 + 130 × 160 + 120 × 100 + 125 × 140 + 130 × 150 + 100 × 120) cm\(^{2}\)
= (16000 + 20800 + 12000 + 17500 + 19500 + 12000) cm\(^{2}\)
= 97800 cm\(^{2}\)
= 9.78 m\(^{2}\)
2. In a square lawn of side 8 m, an N-shaped path is made, as shown in the figure. Find the area of the path.
Solution:
Required area = area of the rectangle PQRS + area of the parallelogram XRYJ + area of the rectangle JKLM
= (2 × 8 + PC × BE + 2 × 8) m\(^{2}\)
= (16 + 2 × 4 + 16) cm\(^{2}\)
= 40 m\(^{2}\)
We can solve this problem using another method:
Required area = Area of the square PSLK – Area of the ∆RYM – Area of the ∆XQJ
= [8 × 8 - \(\frac{1}{2}\){8 – (2 + 2)} × 6 - \(\frac{1}{2}\){8 – (2 + 2)} × 6] m\(^{2}\)
= (64 – 12 – 12) m\(^{2}\)
= 40 m\(^{2}\)
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