Question

for a parallelogram whose sides are 26 cm and 23 cm, which of the following is true? complete parts (a) through (d) below. a. the data are insufficient to enable us to determine the area. choose the correct answer below. a. the statement is false. the area of a parallelogram is equal to the product of its side lengths. b. the statement is false. the area of a parallelogram is equal to the sum of its side lengths. c. the statement is true. the height is unknown, but it is needed to determine the area. d. the statement is true. the exact shape is not known, which is needed to determine which formula for the area to use. b. the area has to be 598 cm². choose the correct answer below. a. the statement is true. the area of a parallelogram is always equal to the product of its side lengths. b. the statement is false. one of the side lengths given is not necessarily the height of the parallelogram, so the area is not necessarily the product of the two values. c. the statement is true. one of the side lengths given is the height of the parallelogram, so its area is the product of the height and the other side length. d. the statement is false. the area of the parallelogram is never equal to the product of its two sides, as the height of the parallelogram is not equal to the side lengths.

Explanation:

Step1: Recall the formula for the area of a parallelogram

The area formula of a parallelogram is \(A = base\times height\). We are given the lengths of the sides (\(a = 26\) cm and \(b=23\) cm), but we don't know the height corresponding to either base.

Step2: Analyze option (a)

• Option A: The area of a parallelogram is not \(side_1\times side_2\). The formula \(A = base\times height\) is different. So, this option is false.
• Option B: The sum of side lengths (\(side_1 + side_2\)) has nothing to do with the area formula. The area formula is \(A=base\times height\). So, this option is false.
• Option C: Since \(A = base\times height\), and we only know the base lengths (not the height), the data is insufficient. This option is true.
• Option D: The formula for the area of a parallelogram is \(A = base\times height\) regardless of the "exact shape" (as long as it is a parallelogram). We just need a base and its corresponding height. So, this option is false.

Step3: Analyze option (b)

• Option A: The area of a parallelogram is not always \(side_1\times side_2\). Since \(A = base\times height\) and height is not necessarily equal to a side length. So, this option is false.
• Option B: Given \(A = base\times height\), if we assume \(base = 26\) cm, we don't know if \(height = 23\) cm (or vice - versa). So, \(26\times23\) (which is \(598\) \(cm^{2}\)) is not necessarily the area. This option is true.
• Option C: We are not given any information that one of the side lengths is the height. So, we can't assume that. This option is false.
• Option D: If the parallelogram is a rectangle (a special case of a parallelogram), then \(height = side\) (since all angles are \(90^{\circ}\)), and \(A=base\times height=side_1\times side_2\). So, the statement "the area of the parallelogram is never equal to the product of its two sides" is false.

Answer:

a. C. The statement is true. The height is unknown, but it is needed to determine the area.
b. B. The statement is false. One of the side lengths given is not necessarily the height of the parallelogram, so the area is not necessarily the product of the two values.