ABCD is a parallelogram with \(\angle ABC = 150^\circ\) and the sides have integer values. If the area of the parallelogram is \(17.5\text{ cm}^2\), then consider the following statements: I. It is possible to have a perimeter equal to 24 cm. II. It is possible to have a perimeter equal to 72 cm. Which of the statements given above is/are correct?
Both I and II
Area of a parallelogram with adjacent sides \(a\) and \(b\) and included angle \(\angle ABC\) is \(ab\sin(\angle ABC)\). Since \(\sin 150^\circ = \tfrac{1}{2}\), \(ab \times \tfrac{1}{2} = 17.5 \Rightarrow ab = 35\). As a, b are positive integers, the factor pairs of 35 are \((1,35)\) and \((5,7)\). For \((5,7)\), perimeter \(=2(5+7)=24\text{ cm}\); for \((1,35)\), perimeter \(=2(1+35)=72\text{ cm}\). Both perimeter values are achievable, so both statements I and II are correct.
Similar Questions
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Important Questions from 2-D Mensuration
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Find its perimeter, when $x = 1$ and $y = -1$. - The ratio between the perimeter and breadth of a rectangle is 3: 1. If the area of the rectangle is $98 \text{ cm}^2$, find the perimeter (in cm) of the rectangle.
The shorter side of a rectangle is 15 cm less than the longer side. The numerical value of its area is equal to 5 times the numerical value of its perimeter. What is the length (in cm) of its longer side?
- In a circle of radius 10.5 cm, if the angle of a sector is $\frac{2\pi}{3}$, then the perimeter of the sector is (in cm):
(Take $\pi = \frac{22}{7}$) - Find the circumference (in m) of the largest circle that can be inscribed in a rectangle whose dimensions are given as 114 m and 63 m.
Take $\pi = \frac{22}{7}$
