Question

A rectangle has a perimeter of 28 units, an area of 48 square units, and sides that are either horizontal or vertical. If one vertex is the point (−5, −7) and the origin is in the interior of the rectangle, find the vertex of the rectangle that is opposite (−5, −7).

Answer 1

To find the vertex of the rectangle that is opposite (-5, -7), we can use the perimeter and area of the rectangle to form equations. By solving these equations, we can find the possible dimensions of the rectangle. The dimensions are either 4 units by 12 units or 12 units by 4 units.

Step-by-step explanation:

To find the vertex of the rectangle that is opposite (-5, -7), we need to analyze the given information. The perimeter of the rectangle is 28 units, which means that each side length is 28/4 = 7 units. The area of the rectangle is 48 square units, so the product of the length and width is 48.

Let's assume the length of the rectangle is x and the width is y. Based on the given information, we have the following equations:

2(x + y) = 28 (perimeter equation)

x * y = 48 (area equation)

To solve these equations, we can substitute y = 48/x into the perimeter equation:

2(x + 48/x) = 28

Multiplying by x to get rid of the fraction:

2x^2 + 96 = 28x

Rearranging the equation:

2x^2 - 28x + 96 = 0

Factoring the quadratic equation:

(x - 4)(2x - 24) = 0

Solving for x:

x = 4 or x = 12

If x = 4, then y = 48/4 = 12. If x = 12, then y = 48/12 = 4.

Therefore, the possible dimensions for the rectangle are 4 units by 12 units or 12 units by 4 units.

Answer 2

Opposite coordinate of (−5, −7) = (1,1)

Step-by-step explanation:

Let the sides of rectangle be a and b

Perimeter = 2 ( a+b ) = 28

a + b = 14

Area = ab = 48

The sides with sum 14 and product 48 is 8 and 6.

a = 8 and b = 6

Since origin is inside the circle, the side with 8 unit is parallel to y axis and side with 6 unit is parallel to x axis

Opposite coordinate of (−5, −7) = (−5+6, −7+8) =(1,1)