Question

a rectangle has a perimeter of 28 units ,an are of 48 square units,and sides that are either horizontal or vertical if on vertex is the point (-5,-7 and the origin is in the interior of the rectangle ,find the vertex of the rectangle that is oppisite( -5,-7)

Answer 1

The opposite vertex to (-5, -7) is (-1, 3).

The perimeter is given as 28 units, so we have the equation:

2a + 2b = 28

Solving for (a) we get:

a = 14 - b

Now, the area is given as 48 square units:

A = ab = (14 - b)b = 48

Solving for (b), we get two possible solutions: (b = 4) or (b = 10).

If (b = 4), then (a = 14 - b = 10).

If (b = 10), then (a = 14 - b = 4).

So, we have two possible rectangles:

1. Dimensions: (a = 10), (b = 4)

2. Dimensions: (a = 4), (b = 10)

Now, let's check which one satisfies the condition that (-5, -7) is one vertex and the origin is in the interior of the rectangle.

For the point (-5, -7):

1. If (a = 10) and (b = 4), the opposite vertex is (-5 + 10, -7 + 4) = (5, -3)

2. If (a = 4) and (b = 10), the opposite vertex is (-5 + 4, -7 + 10) = (-1, 3)

The rectangle with dimensions (a = 4) and (b = 10) satisfies the condition.

Therefore, the opposite vertex to (-5, -7) is (-1, 3).