Question

A circle is plotted on a coordinate plane.
Which equation is the standard form of the equation of a circle drawn on the coordinate plane, derived from the Pythagorean Theorem?

(1 point)
Responses

(x+4)2+(y+3)2=25
left parenthesis x plus 4 right parenthesis squared plus left parenthesis y plus 3 right parenthesis squared equals 25

(x−4)2+(y−3)2=25
left parenthesis x minus 4 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25

x2+(y−5)2=25
x squared plus left parenthesis y minus 5 right parenthesis squared equals 25

x2+y2=25

Answer 1

The standard form of the equation of a circle on a coordinate plane is given by the formula:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

Let's analyze the given options based on this standard form:

1. \((x + 4)^2 + (y + 3)^2 = 25\)
- Center: \((-4, -3)\), Radius: 5 (since \(25 = 5^2\))

2. \((x - 4)^2 + (y - 3)^2 = 25\)
- Center: \((4, 3)\), Radius: 5

3. \(x^2 + (y - 5)^2 = 25\)
- Center: \((0, 5)\), Radius: 5

4. \(x^2 + y^2 = 25\)
- Center: \((0, 0)\), Radius: 5

All of these equations represent circles. However, the first option \((x + 4)^2 + (y + 3)^2 = 25\) and the second option \((x - 4)^2 + (y - 3)^2 = 25\) are explicit representations of the center. The third and fourth options represent circles with the center at the origin or on the y-axis.

In terms of a direct derivation from the Pythagorean Theorem, any option is correct as they all represent a circle.

Therefore, the correct answer in accordance with the standard form and derivation principle would be any of the options, but if asked to select one unambiguously, I would suggest:

\((x + 4)^2 + (y + 3)^2 = 25\)

If that is not the specific option intended to select, you may choose any of the provided options as they conform.