The area of a quarter circle is 19.625 square kilometers. what is the quarter circle's perimeter?

a quarter of a circle is 19.625

 so the area of full circle is 19.625 x 4 = 78.5 sq. km

area of a circle is PI x r^2

78.5 = 3.14 x r^2

78.5/3.14 = 25 = r^2

r = sqrt(25) = 5

radius is 5 km

circumference ( perimeter) = 2 x PI x r

2 x 3.14 x 5 = 31.4 km

 perimeter is 31.4 km

The coordinates of the historic village is (-4 1/2, 5 1/4). The coordinate of the science center is (-4 1/2, -1 3/4). Halfway between the historic village and the science center is given by ((-4 1/2 + -4 1/2) / 2, (5 1/4 + -1 3/4) / 2) = (-9 / 2, 3 1/2 / 2) = (-4 1/2, 1 3/4)
Therefore, the coordinate of the planetarium is (-4 1/2, 1 3/4).