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A circle centered at [tex]$(-1,2)$[/tex] has a diameter of 10 units. Amit wants to determine whether [tex]$(2,-2)$[/tex] is also on the circle. His work is shown below.

The radius is 5 units. Find the distance from the center to [tex]$(2,-2)$[/tex].

[tex]\[
\sqrt{(-1-2)^2 + (2-(-2))^2}
\][/tex]

[tex]\[
\sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]

The point [tex]$(2,-2)$[/tex] lies on the circle because the calculated distance is the same as the radius. Is Amit's work correct?

A. No, he should have used the origin as the center of the circle.
B. No, the radius is 10 units, not 5 units.
C. No, he did not calculate the distance correctly.
D. Yes, the distance from the center to [tex]$(2,-2)$[/tex] is the same as the radius.

Sagot :

GinnyAnswer
To determine whether the point [tex]\((2, -2)\)[/tex] lies on the circle centered at [tex]\((-1, 2)\)[/tex] with a diameter of 10 units, let's go through the necessary steps to verify Amit's claims and calculations.

First, given that the diameter of the circle is 10 units, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{10}{2} = 5 \text{ units} \][/tex]

Next, we need to calculate the distance between the center of the circle [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Applying the coordinates of the center [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]:
[tex]\[ d = \sqrt{(2 - (-1))^2 + (-2 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{(2 + 1)^2 + (-2 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{3^2 + (-4)^2} \][/tex]
[tex]\[ d = \sqrt{9 + 16} \][/tex]
[tex]\[ d = \sqrt{25} \][/tex]
[tex]\[ d = 5 \text{ units} \][/tex]

Since the calculated distance (5 units) is equal to the radius of the circle (5 units), [tex]\((2, -2)\)[/tex] does indeed lie on the circle.

Let's now analyze Amit's work and the statements:
Amit initially tried to calculate the distance:
[tex]\[ \sqrt{(-1 - 2)^2 + (2 - (-2))^2} \][/tex]
But incorrectly calculated the next step:
[tex]\[ \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]

Thus, the correct calculation leads to a distance of 5 units, which matches the radius. This confirms that Amit's final conclusion that the point [tex]\((2, -2)\)[/tex] is not on the circle is incorrect.

Given the information:
- No, he should have used the origin as the center of the circle. (Incorrect, the center is correctly used as [tex]\((-1, 2)\)[/tex]).
- No, the radius is 10 units, not 5 units. (Incorrect, the radius is 5 units).
- No, he did not calculate the distance correctly. (Correct, as explained, Amit's calculations were incorrect).
- Yes, the distance from the center to [tex]\((2,-2)\)[/tex] is not the same as the radius. (Incorrect, the distance is indeed the same as the radius).

Therefore, the correct statement is:
[tex]\[ \text{No, he did not calculate the distance correctly.} \][/tex]
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