Which explains how to find the radius of a circle whose equation is in the form $x^2+y^2=z$?
A. The radius is the constant term, $z$.
B. The radius is the constant term, $z$, divided by 2.
C. The radius is the square root of the constant term, $z$.
D. The radius is the square of the constant term, $z$.
Answer (1)
The equation of a circle centered at the origin is x 2 + y 2 = r 2 .
Given the equation x 2 + y 2 = z , we equate z to r 2 .
Solve for r by taking the square root: r = z .
The radius of the circle is z .
Explanation
Analyze the equation of a circle Let's analyze the equation of a circle centered at the origin, which is given by x 2 + y 2 = r 2 , where r represents the radius of the circle. In our problem, we have the equation x 2 + y 2 = z . We need to find the relationship between z and the radius r .
Compare the given equation with the standard equation Comparing the given equation x 2 + y 2 = z with the standard equation x 2 + y 2 = r 2 , we can see that z corresponds to r 2 . Therefore, we have the equation z = r 2 .
Solve for the radius To find the radius r , we need to take the square root of both sides of the equation z = r 2 . This gives us r = z . Therefore, the radius of the circle is the square root of the constant term z .
Examples
Imagine you're designing a circular garden and you know the area you want the garden to cover. If the equation representing the garden's boundary is x 2 + y 2 = 25 , then z = 25 . To find the radius of the garden, you would take the square root of 25, which is 5. So, the radius of your circular garden would be 5 units (e.g., meters). This helps you determine how much fencing you need or how to arrange plants around the center.
