Question

the area of parallelogram wxyz is approximately 45 square units.
trigonometric area formula: area = \\(\frac{1}{2}ab\sin(c)\\)
what is the approximate perimeter of the parallelogram?
\\(\circ\\) 5 units
\\(\circ\\) 15 units
\\(\circ\\) 30 units
\\(\circ\\) 40 units

Explanation:

Step1: Recall parallelogram area formula

A parallelogram can be divided into two congruent triangles. So the area of parallelogram \( WXYZ \) is \( 2\times\frac{1}{2}ab\sin(C)=ab\sin(C) \), where \( a \) and \( b \) are sides of the triangle (sides of the parallelogram), and \( C \) is the included angle. From the diagram, \( ZY = 10 \), let \( WZ = x \), angle at \( Z \) is \( 115^\circ \), and area of parallelogram is 45. So \( 10\times x\times\sin(115^\circ)=45 \).

Step2: Solve for \( x \)

First, find \( \sin(115^\circ) \). \( \sin(115^\circ)=\sin(90^\circ + 25^\circ)=\cos(25^\circ)\approx0.9063 \). Then the equation becomes \( 10x\times0.9063 = 45 \). So \( x=\frac{45}{10\times0.9063}=\frac{45}{9.063}\approx5 \).

Step3: Calculate perimeter of parallelogram

In a parallelogram, opposite sides are equal. So sides are \( 10 \) and \( 5 \). Perimeter \( P = 2\times(10 + 5)=2\times15 = 30 \) units.

Answer:

30 units (corresponding to the option "30 units")