Question

for a parallelogram whose sides are 26 cm and 23 cm, which of the following is true? complete parts (a) through (d) below.
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 length
c. the area is greater than 598 cm². choose the correct answer below.
a. the statement is false. the area of a parallelogram is always equal to the product of its side lengths.
b. the statement is true. the area of a parallelogram is equal to twice the product of its side lengths
c. the statement is false. the maximum area of the parallelogram is equal to the product of its side lengths, depending on the exact shape
d. this cannot be determined because the height is unknown. the area is greater than or equal to 598 cm²

Explanation:

Step1: Recall the area formula of a parallelogram

The area formula of a parallelogram is \(A = base\times height\). Let the sides of the parallelogram be \(a = 26\) cm and \(b=23\) cm. If we assume the base \(b = 23\) cm, then \(A=23\times h\), where \(h\) is the height corresponding to the base \(23\) cm. Also, using the other side as the base, if \(a = 26\) cm is the base, then \(A = 26\times h'\)

Step2: Analyze part (b)

The product of the side - lengths \(26\times23=598\) cm². But the formula for the area of a parallelogram is \(A=\text{base}\times\text{height}\), not \(\text{side}_1\times\text{side}_2\) unless the parallelogram is a rectangle (where height is equal to the adjacent side). Since we don't know if it is a rectangle (i.e., one of the side - lengths is the height), the area is not necessarily \(26\times23\). So the statement “The area has to be \(598\) cm²” is false.

Step3: Analyze part (c)

The maximum value of the area of a parallelogram with sides \(a\) and \(b\) occurs when it is a rectangle (because \(h\leqslant a\) and \(h'\leqslant b\) depending on the base chosen). Using the formula \(A = base\times height\), if we consider the two sides \(a = 26\) and \(b = 23\) as base and height (in the case of a rectangle), the area \(A=26\times23 = 598\) cm². The area of a non - rectangular parallelogram with sides \(26\) and \(23\) is \(A=\text{base}\times\text{height}\), and \(\text{height}<\text{non - base side}\) (by the property of right - triangles formed in a parallelogram). So the area of the parallelogram is at most \(598\) cm²

Answer:

For part (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
For part (c): C. The statement is false. The maximum area of the parallelogram is equal to the product of its side lengths, depending on the exact shape