Question

Consider a parallelogram in which one side is 3 inches long, another side measures 4 inches, and the measurement of one angle is 45°. How many parallelograms can you construct given these conditions? What are the lengths of the sides and the measurements of the angles for the parallelogram(s)? Using the given information, can you determine the lengths of all the sides of the parallelogram? If so, what are the side lengths?

Answer 1

9514 1404 393

Answer:

  (a) one parallelogram

  (b) opposite sides are 3 inches and 4 inches. Opposite angles are 45° and 135°

  (c) yes, all side lengths can be determined, see (b)

Step-by-step explanation:

Opposite sides of a parallelogram are the same length, so if one side is 3 inches, so is the opposite side. Similarly, if one side is 4 inches, so is the opposite side. If sides have different lengths, they must be adjacent sides. The given numbers tell us the lengths of all of the sides.

The 4 inch sides are adjacent to the 3 inch sides. Thus the angle between a 4 inch side and a 3 inch side must be 45°. Opposite angles are congruent, and adjacent angles are supplementary, so specifying one angle specifies them all.

Only one parallelogram can be formed with these sides and angles. (The acute angle can be at the left end or the right end of the long side. This gives rise to two possible congruent orientations of the parallelogram. Because these are congruent, we claim only one parallelogram is possible. Each is a reflection of the other.)