Final answer:
To show that the points (3, 4), (-4, 3), (-4, -3) are equidistant from the origin, we can use the distance formula to calculate the distance between each point and the origin. The same method can be used to show that the points (4, 6), (0, 6), and (4, 0) are equidistant from the point (2, 3).
Step-by-step explanation:
To show that the points (3, 4), (-4, 3), (-4, -3) are equidistant from the origin:
We can use the distance formula to calculate the distance between each point and the origin. The distance formula is given by:
d = sqrt((x - 0)^2 + (y - 0)^2)
Let's calculate the distance for each point:
d1 = sqrt((3 - 0)^2 + (4 - 0)^2) = sqrt(9 + 16) = sqrt(25) = 5
d2 = sqrt((-4 - 0)^2 + (3 - 0)^2) = sqrt(16 + 9) = sqrt(25) = 5
d3 = sqrt((-4 - 0)^2 + (-3 - 0)^2) = sqrt(16 + 9) = sqrt(25) = 5
Since the distance from each point to the origin is 5, we can conclude that the points (3, 4), (-4, 3), (-4, -3) are equidistant from the origin.
To show that the points (4, 6), (0, 6), and (4, 0) are equidistant from the point (2, 3):
We can again use the distance formula to calculate the distance between each point and (2, 3):
d = sqrt((x - 2)^2 + (y - 3)^2)
Let's calculate the distance for each point:
d1 = sqrt((4 - 2)^2 + (6 - 3)^2) = sqrt(4 + 9) = sqrt(13)
d2 = sqrt((0 - 2)^2 + (6 - 3)^2) = sqrt(4 + 9) = sqrt(13)
d3 = sqrt((4 - 2)^2 + (0 - 3)^2) = sqrt(4 + 9) = sqrt(13)
Since the distance from each point to (2, 3) is sqrt(13), we can conclude that the points (4, 6), (0, 6), and (4, 0) are equidistant from the point (2, 3).