Question

3
select the correct answer.
the parallelogram has an area of 20 square inches. what are the dimensions of the parallelogram, to the nearest hundredth of an inch?
image of parallelogram with triangle, angle 40°, side 4 in, height h, base x
hint: use the formula that uses trigonometry to find the area of any non - right triangle in order to solve this problem.
( a_{\text{triangle}} = \frac{1}{2} ab sin(c) )
a. ( x = 4.00 ) in, ( h = 5.00 ) in
b. ( x = 7.78 ) in, ( h = 2.57 ) in
c. ( x = 3.06 ) in, ( h = 6.54 ) in
d. ( x = 6.22 ) in, ( h = 3.23 ) in

Explanation:

Step1: Analyze the parallelogram and triangle relation

A parallelogram can be divided into two congruent triangles. So the area of the parallelogram \( A_{parallelogram} = 2\times A_{triangle} \). Given \( A_{parallelogram}=20 \) square inches, so \( A_{triangle}=\frac{20}{2} = 10 \) square inches. The formula for the area of the triangle is \( A_{triangle}=\frac{1}{2}ab\sin(C) \), here \( a = 4 \) in, \( b=x \), \( C = 40^\circ \), and \( A_{triangle}=10 \).

Step2: Solve for \( x \)

Substitute the values into the triangle area formula: \( 10=\frac{1}{2}\times4\times x\times\sin(40^\circ) \). First, simplify the right - hand side: \( \frac{1}{2}\times4 = 2 \), so the equation becomes \( 10 = 2x\sin(40^\circ) \). We know that \( \sin(40^\circ)\approx0.6428 \), then \( 10=2x\times0.6428 \), \( 2x\times0.6428 = 10 \), \( 1.2856x=10 \), \( x=\frac{10}{1.2856}\approx7.78 \) in.

Step3: Solve for \( h \)

We can also use the area formula of the parallelogram \( A = x\times h \) (since the area of a parallelogram is base times height). We know \( A = 20 \) and \( x\approx7.78 \), so \( h=\frac{20}{x}=\frac{20}{7.78}\approx2.57 \) in. Or we can use trigonometry: in the right - triangle formed by the height \( h \), the side of length 4 in and the angle of \( 40^\circ \), \( \sin(40^\circ)=\frac{h}{4} \), so \( h = 4\sin(40^\circ)\approx4\times0.6428 = 2.57 \) in.

Answer:

B. \( x = 7.78 \) in, \( h = 2.57 \) in