The point (-3,4) is on a circle with its center at the origin. Which of the following points must also be on the circle?

None of your "following points" are showing up

Point with integer values on the circle are

(5,0), (-5,0), (0, 5) , (0, -5)
(3,4) ,(-3,4) , (3, -4), and (-3,-4)

Jodie's point of (0,-4) does not lie on the circle

the equation would be
x^2 + y^2 = 25
so sub in your given points to see if they satisfy that equation.
If they do, they lie on the circle.

Well, since the center of the circle is at the origin (0,0), any other point that is equidistant from the origin must also be on the circle. So, grab your measuring tape and find those points! Just kidding, don't grab a measuring tape, this is math, not DIY. Anyway, all other points that are equally distant from the origin are also on the circle. So, basically, any point that forms a line with the origin and the point (-3,4) will be on the circle. It could be (-6,8) or (-9,12), or any other point you can come up with as long as it's equidistant from the origin. Easy peasy, lemon squeezy!

To determine which points must also be on the circle with the center at the origin, we can use the distance formula.

The distance between the center of the circle (0, 0) and the point (-3, 4) is calculated as:

d = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-3 - 0)^2 + (4 - 0)^2)
= √((-3)^2 + 4^2)
= √(9 + 16)
= √25
= 5

Any point on the circle will have the same distance from the center.

Now, let's check the points given to see if they also have a distance of 5 from the origin:

A) (1, 0)
d = √((1 - 0)^2 + (0 - 0)^2)
= √(1 + 0)
= √1
= 1

B) (0, 5)
d = √((0 - 0)^2 + (5 - 0)^2)
= √(0 + 25)
= √25
= 5

C) (-4, -3)
d = √((-4 - 0)^2 + (-3 - 0)^2)
= √(16 + 9)
= √25
= 5

From the calculations, we can see that point (-3, 4) on the circle with its center at the origin, the points (0, 5) and (-4, -3) must also be on the circle. These points have a distance of 5 from the origin. Therefore, the correct answer is B) (0, 5) and C) (-4, -3).

To determine which points must also be on the circle with center at the origin, we need to find the equation of the circle first.

The equation of a circle with center (h,k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2

Since the center of the circle is at the origin (0,0), the equation of the circle can be simplified to:
x^2 + y^2 = r^2

Now, let's substitute the coordinates of the point (-3,4) and solve for r^2:
(-3)^2 + 4^2 = r^2
9 + 16 = r^2
25 = r^2

Therefore, the equation of the circle with center at the origin is:
x^2 + y^2 = 25

Now that we have the equation of the circle, let's check which of the following points satisfy this equation, thus must also be on the circle:

A) (-4,0):
(-4)^2 + 0^2 = 16 ≠ 25

B) (0,-5):
0^2 + (-5)^2 = 25

C) (3,-4):
3^2 + (-4)^2 = 25

D) (5,12):
5^2 + 12^2 = 169 ≠ 25

From the calculations, we see that the points (0,-5) and (3,-4) satisfy the equation of the circle. Therefore, these points must also be on the circle with center at the origin.

Well, the origin must mean that the coordinates are (0,0), so (-3,4) and (0,-4) are also points on the the circle.