How do i Solve the Radius of the circle?

Question

Hello I was wondering about this kind of problem I'm having. Here it is: $$ x^2 + y^2 = 49 $$

Formula given by our instructor is: $x^2 + y^2 = r^2$.

Answer 1

We know that $x^2+y^2=49$ and we also know that $x^2+y^2=r^2$

Since we know two things that $x^2+y^2$ is equal to, this means that those things must also be equal.

$$r^2=49$$ If we take the square root $$r=7$$

Answer 2

The reason why $x^2 + y^2 = r^2$ gives you a circle centered at the origin of radius $r$ is as follows:

Imagine you are given some point $(x, y)$ in the plane. Now we can draw a right triangle with the hypotenuse being the segment from the origin to $(x, y)$. If we denote the length of the hypotenuse $c$, then by the Pythagorean theorem, we have $c^2 = x^2 + y^2$.

Now suppose you wanted to describe the set of all points that are distance $c$ from the origin. Well, they all satisfy the same equation: $x^2 + y^2 = c^2$ from the Pythagorean theorem, and that set will trace out a circle of radius $c$.

To conclude, if you are given an equation for the circle $x^2 + y^2 = r^2$, then it's radius is simply $+\sqrt{r^2} = r$.