Question

what is the diameter of this circle?

Explanation:

Step1: Find the radius

The center of the circle is at \((-2, 0)\) (from the green dot). The circle intersects the \(x\)-axis at \(x = 5\) (rightmost point) and \(x=-9\) (leftmost point)? Wait, no, looking at the grid, the rightmost point on the circle is at \(x = 5\)? Wait, no, the grid lines: each grid square is 1 unit. The center is at \((-2, 0)\). The rightmost point of the circle is at \(x = 5\)? Wait, no, let's check the horizontal distance from center to rightmost point. The center is at \(x=-2\), and the rightmost point is at \(x = 5\)? Wait, no, looking at the graph, the circle goes from \(x=-9\) to \(x = 5\)? Wait, no, maybe I miscalculate. Wait, the center is at \((-2, 0)\). Let's find the distance from center to the rightmost point on the \(x\)-axis. The rightmost point where the circle meets the \(x\)-axis is at \(x = 5\)? Wait, no, the grid: from \(x=-2\) (center) to \(x = 5\) is \(5 - (-2)=7\)? Wait, no, maybe the rightmost point is at \(x = 5\)? Wait, no, looking at the graph, the circle's rightmost point is at \(x = 5\) (since the grid line at \(x=5\) is where the circle touches the \(x\)-axis on the right). Wait, no, the center is at \((-2, 0)\), and the leftmost point is at \(x=-9\)? Wait, no, let's count the grid squares. From the center \((-2, 0)\) to the rightmost point on the circle (where it intersects the \(x\)-axis on the right) is how many units? Let's see, the center is at \(x=-2\), and the rightmost point is at \(x = 5\)? Wait, no, the grid lines: each square is 1 unit. So from \(x=-2\) to \(x = 5\) is \(5 - (-2)=7\)? Wait, no, maybe I made a mistake. Wait, the circle's diameter is the distance between the leftmost and rightmost points. Let's find the leftmost \(x\)-coordinate and rightmost \(x\)-coordinate of the circle. Looking at the graph, the leftmost point is at \(x=-9\) (since the circle touches the \(x\)-axis at \(x=-9\)) and the rightmost point is at \(x = 5\)? Wait, no, that can't be. Wait, maybe the center is at \((-2, 0)\), and the radius is the distance from center to the rightmost point. Let's check the \(x\)-coordinate of the rightmost point: looking at the grid, the circle goes to \(x = 5\) (since the vertical line at \(x=5\) is where the circle meets the \(x\)-axis on the right). So the distance from center \((-2, 0)\) to \(x=5\) is \(5 - (-2)=7\)? Wait, no, that would be radius 7, diameter 14? Wait, no, maybe I miscounted. Wait, let's look again. The center is at \((-2, 0)\). The leftmost point of the circle: let's see, the circle touches the \(x\)-axis at \(x=-9\)? Wait, no, the grid lines: from \(x=-10\) to \(x=10\). The center is at \(x=-2\). The distance from center to the leftmost point: let's count the grid squares. From \(x=-2\) to \(x=-9\) is 7 units? Wait, no, \( -2 - (-9)=7\)? Wait, no, \( -9 - (-2)= -7\), absolute value 7. So radius is 7, diameter is \(2\times7 = 14\)? Wait, but let's check the vertical direction. The topmost point is at \(y = 7\) (since center is at \(y=0\), and the topmost point is at \(y=7\)), so radius is 7, diameter 14. Alternatively, the distance between the leftmost and rightmost points: leftmost \(x=-9\), rightmost \(x=5\), so distance is \(5 - (-9)=14\), which is the diameter. Yes, that makes sense. So the diameter is 14.

Step2: Confirm the diameter

The diameter of a circle is twice the radius, or the distance between two points on the circle passing through the center. Here, the horizontal distance between the leftmost (\(x=-9\)) and rightmost (\(x=5\)) points is \(5 - (-9)=14\), so the diameter is 14.

Answer:

14