Question

what is the radius of this circle?

Explanation:

Step1: Identify center and a point on the circle

The center of the circle is at \((0, 9)\) (since it's on the y - axis, x - coordinate is 0 and y - coordinate is 9). A point on the circle is at \((0, 8)\) (or we can also use the top point \((0,10)\)). The radius is the distance between the center \((0,9)\) and a point on the circle, say \((0,8)\) or \((0,10)\).

Step2: Calculate the radius

Using the distance formula for points on the same vertical line (\(x\) - coordinates are the same), the distance \(d=\vert y_2 - y_1\vert\). If we take center \((0,9)\) and point \((0,10)\), then \(d=\vert10 - 9\vert= 1\)? Wait, no, wait. Wait, looking at the grid, the center is at \((0,9)\) (since from \(y = 8\) to \(y=10\), the center is in the middle? Wait, no, the circle is between \(y = 8\) and \(y = 10\) on the y - axis. So the center is at \((0,9)\), and the top of the circle is at \((0,10)\), bottom at \((0,8)\). So the radius is the distance from center \((0,9)\) to \((0,10)\), which is \(10 - 9=1\)? Wait, no, that can't be. Wait, maybe the center is at \((0,9)\) and the circle extends from \(y = 8\) to \(y = 10\), so the diameter is \(10 - 8 = 2\), so radius is \(\frac{2}{2}=1\)? Wait, no, wait the grid lines: each grid square is 1 unit. The center is at \((0,9)\), and the circle goes from \(x=- 1\) to \(x = 1\) (since it's a circle on the grid) and \(y = 8\) to \(y = 10\). Wait, maybe I misread the center. Wait, the blue dot is the center. Let's check the coordinates. The y - axis is the vertical line \(x = 0\). The center is at \((0,9)\) (since it's on the y - axis, and between \(y = 8\) and \(y = 10\)). The circle touches \(y = 8\) and \(y = 10\), so the distance from center \((0,9)\) to \(y = 8\) is \(9 - 8=1\), and to \(y = 10\) is \(10 - 9 = 1\). Also, horizontally, it touches \(x=-1\) and \(x = 1\), so distance from center \((0,9)\) to \(x=-1\) (or \(x = 1\)) is \(1-0 = 1\) (since x - coordinate of center is 0). So the radius is 1? Wait, no, that seems too small. Wait, maybe the center is at \((0,9)\) and the circle has a radius of 1? Wait, let's count the grid squares. From the center (blue dot) to the edge of the circle: if we move up 1 unit, we reach the top of the circle, move down 1 unit, reach the bottom, move left 1 unit, reach the left edge, move right 1 unit, reach the right edge. So the radius is 1. Wait, but maybe I made a mistake. Wait, the circle is drawn with center at \((0,9)\) and radius 1, because the distance from center to any point on the circle is 1. So the radius is 1.

Wait, no, wait the grid: each square is 1 unit. The center is at (0,9), and the circle goes from x=-1 to x=1 (so width 2, radius 1) and y=8 to y=10 (height 2, radius 1). So the radius is 1.

Answer:

1