Question

a circle centered at (-1, 2) has a diameter of 10 units. amit wants to determine whether (2, -2) is also on the circle. his work is shown below. the radius is 5 units. find the distance from the center to (2, -2). $sqrt{(-1 - 2)^2+(2-(-2))^2}$ $sqrt{(-3)^2+(4)^2}=3$. the point (2, -2) doesnt lie on the circle because the calculated distance should be the same as the radius. is amits work correct? no, he should have used the origin as the center of the circle. no, the radius is 10 units, not 5 units. no, he did not calculate the distance correctly. yes, the distance from the center to (2, -2) is not the same as the radius.

Explanation:

Step1: Calculate the correct distance

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, the center of the circle is $(x_1,y_1)=(-1,2)$ and the point is $(x_2,y_2)=(2,-2)$. So $d=\sqrt{(2 - (-1))^2+(-2 - 2)^2}=\sqrt{(2 + 1)^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step2: Analyze Amit's work

Amit made an error in calculating $\sqrt{(-1 - 2)^2+(2-(-2))^2}$. He incorrectly calculated $\sqrt{(-3)^2+(4)^2}$ as 3. The correct value is $\sqrt{9 + 16}=5$. The radius is 5 units (since diameter = 10 units, radius=$\frac{10}{2}=5$ units).

Answer:

No, he did not calculate the distance correctly