Questions: The set of points in a plane that are the same distance from a fixed point is called a circle. The standard form of the equation of the circle whose center is at the origin may be derived using the distance formula. The Pythagorean Theorem can be used to derive the distance formula. The center of the circle is the midpoint of its diameter.

Solution Steps

5. The set of points in a plane that are the same distance from a fixed point is called a circle.

6. The standard form of the equation of the circle whose center is at the origin may be derived using the distance formula.

7. The Pythagorean Theorem can be used to derive the distance formula.

8. The center of the circle is the midpoint of its diameter.

To find the distance between the points \( (0, 0) \) and \( (3, 4) \), we use the distance formula:

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

Substituting the coordinates:

\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0 \]

To find the midpoint of the line segment connecting the points \( (0, 0) \) and \( (3, 4) \), we use the midpoint formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates:

\[ M = \left( \frac{0 + 3}{2}, \frac{0 + 4}{2} \right) = \left( \frac{3}{2}, 2 \right) = (1.5, 2.0) \]

Final Answer