Question

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, 40°, 4 in, x, h
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 \( A_{triangle}=\frac{1}{2}ab\sin(C) \), then \( A_{parallelogram}=ab\sin(C) \). In the diagram, one side of the triangle is \( x \), another is \( 4 \) in, and the included angle is \( 40^\circ \), and also \( h = 4\sin(40^\circ) \) (since \( h \) is the height related to the angle in the right - triangle formed).

Step2: Calculate the height \( h \)

We know that in the right - triangle with hypotenuse \( 4 \) in and angle \( 40^\circ \), \( h = 4\sin(40^\circ) \). Using a calculator, \( \sin(40^\circ)\approx0.6428 \), so \( h = 4\times0.6428 = 2.5712\approx2.57 \) in.

Step3: Calculate the base \( x \)

The area of the parallelogram \( A = x\times h=20 \). We know \( h\approx2.57 \) in, so \( x=\frac{20}{h} \). Substituting \( h = 2.57 \) (or we can use the formula \( A = x\times4\sin(40^\circ) \), so \( x=\frac{20}{4\sin(40^\circ)}=\frac{5}{\sin(40^\circ)} \)). Since \( \sin(40^\circ)\approx0.6428 \), \( x=\frac{5}{0.6428}\approx7.78 \) in.

Answer:

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