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 $(2,-2)$.
[tex]$\sqrt{(-1-2)^2+(2-(-2))^2}$[/tex]
[tex]$\sqrt{(-3)^2+(0)^2}=3$[/tex]
The point $(2,-2)$ doesn't lie on the circle because the calculated distance should be 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 $(2,-2)$ is not the same as the radius.
Answer (2)
Calculate the distance between the center of the circle ( − 1 , 2 ) and the point ( 2 , − 2 ) using the distance formula: d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2 .
Simplify the expression: d = ( 3 ) 2 + ( − 4 ) 2 = 9 + 16 = 25 .
Find the square root: d = 5 .
Conclude that Amit's work is incorrect because he did not calculate the distance correctly, and the point ( 2 , − 2 ) lies on the circle since the distance equals the radius: No, he did not calculate the distance correctly.
Explanation
Analyze Amit's work The problem is to determine if Amit's calculation and conclusion about whether the point (2, -2) lies on the circle centered at (-1, 2) with a diameter of 10 is correct. Amit's calculation of the distance between the center and the point (2, -2) is incorrect.
State the distance formula To determine if Amit's work is correct, we need to calculate the distance between the center of the circle (-1, 2) and the point (2, -2) using the distance formula:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Substitute the coordinates Substitute the coordinates of the center (-1, 2) and the point (2, -2) into the distance formula:
d = ( 2 − ( − 1 ) ) 2 + ( − 2 − 2 ) 2
Simplify the expression Simplify the expression:
d = ( 2 + 1 ) 2 + ( − 4 ) 2 d = ( 3 ) 2 + ( − 4 ) 2
Calculate the squares Calculate the squares:
d = 9 + 16 d = 25
Find the square root Find the square root:
d = 5
Compare the distance with the radius The calculated distance between the center of the circle and the point (2, -2) is 5 units. The radius of the circle is also 5 units (since the diameter is 10 units). Since the distance between the center and the point is equal to the radius, the point (2, -2) lies on the circle.
Identify Amit's error Amit's calculation was ( − 1 − 2 ) 2 + ( 2 − ( − 2 ) ) 2 = ( − 3 ) 2 + ( 0 ) 2 = 3 . This is incorrect because he made an error in calculating the y-component of the distance. The correct distance is 5.
Conclusion Amit's work is incorrect because he did not calculate the distance correctly. The correct distance is 5, which is equal to the radius of the circle. Therefore, the point (2, -2) lies on the circle.
Final Answer The correct answer is: No, he did not calculate the distance correctly.
Examples
Understanding circles and distances is crucial in many real-world applications. For example, in GPS navigation, your phone calculates its distance from several satellites to determine your location. Each satellite's signal represents a circle (or sphere in 3D), and your phone lies at the intersection of these circles. This principle is also used in surveying, astronomy, and even in designing wireless networks to ensure optimal coverage.
Amit's calculation of the distance from the center of the circle to the point ( 2 , − 2 ) was incorrect. The correct distance is 5 units, which is equal to the radius of the circle. Thus, the point ( 2 , − 2 ) does lie on the circle.
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