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 the square root of the constant term z : 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 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
State the final answer 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 equation representing its boundary is x 2 + y 2 = 25 . To determine how much fencing you need (which depends on the radius), you take the square root of 25, which gives you a radius of 5 units. This tells you the garden extends 5 units in all directions from the center, helping you plan your garden layout and fencing needs effectively.
