Find the perimeter and area of the rectangle with vertices −1,  2, 1,  7, −3,  7, and −−3,  2.

To find the perimeter and area of the rectangle, we first need to determine the lengths of its sides. The vertices of the rectangle are given as (-1,2), (1,7), (-3,7), and (-3,2).

Step 1: Calculate the length of the sides We can use the distance formula to calculate the lengths of the sides. The distance formula is √[(x₂ - x₁)² + (y₂ - y₁)²].

For the side between (-1,2) and (1,7), the length is √[(1 - (-1))² + (7 - 2)²] = √[(2)² + (5)²] = √[4 + 25] = √29.

For the side between (-1,2) and (-3,2), the length is √[(-3 - (-1))² + (2 - 2)²] = √[(-2)² + (0)²] = √[4 + 0] = √4 = 2.

Step 2: Calculate the perimeter The perimeter of a rectangle is given by the formula 2(length + width). So, the perimeter is 2(√29 + 2) = 2√29 + 4.

Step 3: Calculate the area The area of a rectangle is given by the formula length * width. So, the area is √29 * 2 = 2√29.

So, the perimeter of the rectangle is 2√29 + 4 and the area is 2√29.