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The distance formula is a tool used in geometry to calculate the distance between two points in a Cartesian coordinate system. It's derived from the Pythagorean theorem and is especially helpful when you need to find the distance between any two points with given coordinates. The formula is:

• \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In the exercise, the points given are \(9, -4\) and \(-3,5\). Here, \(x_1, y_1\) represents the first point \(-3, 5\) (the circle's center), and \(x_2, y_2\) is the point on the circle. By substituting these values into the formula, you can calculate the distance, which in this context equals the length of the radius of the circle, as it stretches from the center to the perimeter. This is an essential step in determining the circle's area.

The radius of a circle is any line segment from the center of the circle to its perimeter. Calculating the radius is crucial because it is used in various formulas, including the area of a circle. In this exercise, you're asked to use the distance formula, which we just covered, to compute the radius.

After substituting the coordinates into the distance formula \(\sqrt{(9 - (-3))^2 + ((-4) - 5)^2}\), you'll need to simplify each part:

• Subtract the x-coordinates: \((9 - (-3)) \rightarrow 9 + 3 = 12\)
• Subtract the y-coordinates: \((-4) - 5 \rightarrow -4 - 5 = -9\)
• Then, square both results: \(12^2 = 144\) and \((-9)^2 = 81\)
• Add the squares and take the square root: \(\sqrt{144 + 81} = \sqrt{225} = 15\)

The calculated radius of the circle is 15 units. This value is key to finding the circle's area by substituting it back into the appropriate area formula.

Circle geometry involves understanding the properties and formulas related to circles. One of the primary properties in circle geometry is the area. The area of a circle is the space contained within its circumference. To calculate the area, you use the formula:

• \[ \text{Area} = \pi r^2 \]

The symbol \(\pi\) represents the mathematical constant pi, approximately equal to 3.14159. It is the ratio of a circle's circumference to its diameter.

In the exercise, after finding the radius \( r = 15 \), substitute it into the area formula:

• \[\text{Area} = \pi \times 15^2 = \pi \times 225\]

Multiplying through gives the circle's exact area \(225\pi\) square units. This formula not only helps in solving geometry problems but also deepens understanding of how different parts of a circle's geometry coexist.