Question

what is the radius of this circle?

Explanation:

Step1: Identify the center and a point on the circle

The center of the circle is at \((0, -4)\) (from the graph, the yellow dot). A point on the circle along the y - axis is \((0, -2)\) or \((0, -6)\). Let's take the center \((x_1,y_1)=(0, - 4)\) and a point on the circle \((x_2,y_2)=(0, - 2)\).

Step2: Calculate the distance between the center and the point (which is the radius)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Since \(x_1 = x_2=0\), the formula simplifies to \(d=\vert y_2 - y_1\vert\).

Substitute \(y_1=-4\) and \(y_2 = - 2\) into the formula: \(d=\vert-2-(-4)\vert=\vert-2 + 4\vert=\vert2\vert = 2\). We can also check with the other point \((0,-6)\): \(d=\vert-6-(-4)\vert=\vert-6 + 4\vert=\vert-2\vert = 2\).

Answer:

The radius of the circle is 2.