Question

question 21 of 39
the proof that point (1, √3) lies on the circle that is centered at the origin and contains the point (0, 2) is found in the table below. what is the justification for the 2nd statement?
a circle is centered at (0, 0) and contains the point (0, 2). given
the radius of the circle is the distance from (0, 0) to (0, 2).
the distance from (0, 0) to (0, 2) is √(0 - 0)²+(2 - 0)² = √2² = 2 distance formula
if (1, √3) lies on the circle it must be the same distance from the center as (0, 2). definition of a circle
the distance from (1, √3) is √(0 - 1)²-(0 - √3)² = √1 + 3 = 2 distance formula
since (1, √3) is 2 units from (0, 0), it lies on a circle that is centered at the origin and contains the point (0, 2). definition of a circle

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Analyze the second statement

The second statement is calculating the distance from \((0,0)\) to \((0,2)\) using the formula \(d=\sqrt{(0 - 0)^2+(2 - 0)^2}\). This is a direct application of the distance formula.

Answer:

Distance formula