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Sagot :
To solve the problem of determining which point lies on a circle with a radius of 5 units and center at P(6, 1), we will use the distance formula to find the distances from the center of the circle to each of the given points. The distance formula is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the center of the circle and [tex]\( (x_2, y_2) \)[/tex] is a point on the plane.
Given:
- Center of the circle: [tex]\( P(6, 1) \)[/tex]
- Radius of the circle: [tex]\( 5 \)[/tex] units
Points to check:
- [tex]\( Q(1, 11) \)[/tex]
- [tex]\( R(2, 4) \)[/tex]
- [tex]\( S(4, -4) \)[/tex]
- [tex]\( T(9, -2) \)[/tex]
We will calculate the distance from the center [tex]\( P \)[/tex] to each of these points.
1. Distance to point Q(1, 11):
[tex]\[ d = \sqrt{(6 - 1)^2 + (1 - 11)^2} \][/tex]
[tex]\[ d = \sqrt{5^2 + (-10)^2} \][/tex]
[tex]\[ d = \sqrt{25 + 100} \][/tex]
[tex]\[ d = \sqrt{125} \][/tex]
[tex]\[ d = 11.18 \][/tex]
2. Distance to point R(2, 4):
[tex]\[ d = \sqrt{(6 - 2)^2 + (1 - 4)^2} \][/tex]
[tex]\[ d = \sqrt{4^2 + (-3)^2} \][/tex]
[tex]\[ d = \sqrt{16 + 9} \][/tex]
[tex]\[ d = \sqrt{25} \][/tex]
[tex]\[ d = 5 \][/tex]
3. Distance to point S(4, -4):
[tex]\[ d = \sqrt{(6 - 4)^2 + (1 - (-4))^2} \][/tex]
[tex]\[ d = \sqrt{2^2 + 5^2} \][/tex]
[tex]\[ d = \sqrt{4 + 25} \][/tex]
[tex]\[ d = \sqrt{29} \][/tex]
[tex]\[ d = 5.39 \][/tex]
4. Distance to point T(9, -2):
[tex]\[ d = \sqrt{(6 - 9)^2 + (1 - (-2))^2} \][/tex]
[tex]\[ d = \sqrt{(-3)^2 + 3^2} \][/tex]
[tex]\[ d = \sqrt{9 + 9} \][/tex]
[tex]\[ d = \sqrt{18} \][/tex]
[tex]\[ d = 4.24 \][/tex]
Next, we compare these distances with the radius of the circle, which is 5 units.
- The distance to point [tex]\( Q(1, 11) \)[/tex] is 11.18 units, which is greater than 5.
- The distance to point [tex]\( R(2, 4) \)[/tex] is 5 units, which matches the radius of the circle.
- The distance to point [tex]\( S(4, -4) \)[/tex] is 5.39 units, which is greater than 5.
- The distance to point [tex]\( T(9, -2) \)[/tex] is 4.24 units, which is less than 5.
Therefore, the point that lies exactly on the circle with a radius of 5 units and center at [tex]\( P(6, 1) \)[/tex] is [tex]\( R(2, 4) \)[/tex].
The correct answer is:
B. R(2, 4)
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the center of the circle and [tex]\( (x_2, y_2) \)[/tex] is a point on the plane.
Given:
- Center of the circle: [tex]\( P(6, 1) \)[/tex]
- Radius of the circle: [tex]\( 5 \)[/tex] units
Points to check:
- [tex]\( Q(1, 11) \)[/tex]
- [tex]\( R(2, 4) \)[/tex]
- [tex]\( S(4, -4) \)[/tex]
- [tex]\( T(9, -2) \)[/tex]
We will calculate the distance from the center [tex]\( P \)[/tex] to each of these points.
1. Distance to point Q(1, 11):
[tex]\[ d = \sqrt{(6 - 1)^2 + (1 - 11)^2} \][/tex]
[tex]\[ d = \sqrt{5^2 + (-10)^2} \][/tex]
[tex]\[ d = \sqrt{25 + 100} \][/tex]
[tex]\[ d = \sqrt{125} \][/tex]
[tex]\[ d = 11.18 \][/tex]
2. Distance to point R(2, 4):
[tex]\[ d = \sqrt{(6 - 2)^2 + (1 - 4)^2} \][/tex]
[tex]\[ d = \sqrt{4^2 + (-3)^2} \][/tex]
[tex]\[ d = \sqrt{16 + 9} \][/tex]
[tex]\[ d = \sqrt{25} \][/tex]
[tex]\[ d = 5 \][/tex]
3. Distance to point S(4, -4):
[tex]\[ d = \sqrt{(6 - 4)^2 + (1 - (-4))^2} \][/tex]
[tex]\[ d = \sqrt{2^2 + 5^2} \][/tex]
[tex]\[ d = \sqrt{4 + 25} \][/tex]
[tex]\[ d = \sqrt{29} \][/tex]
[tex]\[ d = 5.39 \][/tex]
4. Distance to point T(9, -2):
[tex]\[ d = \sqrt{(6 - 9)^2 + (1 - (-2))^2} \][/tex]
[tex]\[ d = \sqrt{(-3)^2 + 3^2} \][/tex]
[tex]\[ d = \sqrt{9 + 9} \][/tex]
[tex]\[ d = \sqrt{18} \][/tex]
[tex]\[ d = 4.24 \][/tex]
Next, we compare these distances with the radius of the circle, which is 5 units.
- The distance to point [tex]\( Q(1, 11) \)[/tex] is 11.18 units, which is greater than 5.
- The distance to point [tex]\( R(2, 4) \)[/tex] is 5 units, which matches the radius of the circle.
- The distance to point [tex]\( S(4, -4) \)[/tex] is 5.39 units, which is greater than 5.
- The distance to point [tex]\( T(9, -2) \)[/tex] is 4.24 units, which is less than 5.
Therefore, the point that lies exactly on the circle with a radius of 5 units and center at [tex]\( P(6, 1) \)[/tex] is [tex]\( R(2, 4) \)[/tex].
The correct answer is:
B. R(2, 4)
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