If the width of the rectangle is 40% of the length, then 

.

The perimeter of the rectangle is:

The perimeter is 210 inches, so we can solve for the length:

The length and width of the rectangle are 75 and 30 inches; the area is their product, or

 square inches.

This might be easier to solve if you set .

Then the dimensions of Rectangle A are  and . The area of Rectangle A is their product:

The dimensions of Rectangle B are  and .  The area of Rectangle B is their product:

 regardless of the value of  (or, subsequently, ), so Rectangle A has the greater area.

The area of a rectangle is the product of its length and its width, which here are  and . Mulitply:

The lengths of the diagonals of these rectangles can be computed using the Pythagorean Theorem:

(a) 

(b) 

 so . Since the diagonal of the rectangle in (b) measures less than , it must also measure less than that of the square in (a)

The diagonals of a rectangle are congruent and bisect each other. Therefore,  is equidistant from all four vertices, making . The relationship between the sides is not relevant here.

Two consecutive sides of a rectangle and a diagonal form a right triangle, so the length of any diagonal can be determined using the Pythagorean Theorem, substituting :

The diagonals of a rectangle bisect each other. Therefore, the distance from a vertex to the point of intersection is half this, and .

Let and  be the dimensions of the rectangle. Then 

 and, subsequently, 

Since the product of the length and width is the area, we are looking for two numbers whose sum is 70 and whose product is 1,200; through trial and error, they are found to be 30 and 40. We can assign either to be  and the other to be  since the result is the same.

The length of a diagonal of the rectangle  can be found by applying the Pythagorean Theorem:

A diagonal is 50 inches long; since 4 feet are equivalent to 48 inches, (A) is the greater quantity.

Given that w = 2x and l = 1.5w + 5, a substitution will show that l = 1.5(2x) + 5 = 3x + 5.  

P = 2w + 2l = 2(2x) + 2(3x + 5) = 4x + 6x + 10 = 10x + 10 = 10(x + 1)

The perimeter of a rectangle is equal to twice the sum of its length and its width, which here would be, in inches,

.

Therefore, the correct choice is that all four measurements are equal to the perimeter.

A square with area 10,000 square centimeters has sidelength  centimeters, and perimeter  centimeters. 

Not enough information is given about the rectangle with area 8,000 square centimeters to determine its perimeter. For example, if its dimensions are 100 centimeters by 80 centimeters, its perimeter is  centimeters. If the dimensions are 200 centimeters by 40 centimeters, its perimeter is  centimeters. Both cases are consistent with the conditions of the problem, yet one makes (a) greater and one makes (b) greater.