Question

A rectangle has a length that is a rational number and a width that is an irrational number.
Which TWO of the following statements are correct?
A.
The area of the rectangle is a Rational Number
B.
The Perimeter of the rectangle is an Irrational Number
C.
The Area of the rectangle is an irrational Number
D.
The perimeter of the rectangle is a rational Number
E.
The area and the perimeter for this rectangle are equal

Answer 1

Answer: B and C.

Reason: Let length = r (rational) and width = s (irrational), with r > 0, s > 0.

- Area = r·s. A nonzero rational times an irrational is irrational, so the area is irrational → C true, A false.
- Perimeter = 2(r + s). r + s is irrational (a rational plus an irrational is irrational), so the perimeter is irrational → B true, D false.
- Could area = perimeter? r·s = 2(r + s) implies s = 2r/(r − 2) for r ≠ 2, which is rational, contradicting s irrational; r = 2 gives s = 2 rational. So E is false.