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1 answer:
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Answer:
A. (0, 5), (5, 1)
Step-by-step explanation:
The midpoints of the diagonals of a parallelogram are the same point. This means the sum of the opposite vertex coordinates will be the same.
For the given diagonal, the sum of end point coordinates is ...
(-2, -3) + (7, 7) = (-2+7, -3+7) = (5, 4)
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This means we're looking for a pair of coordinates whose sum is (5, 4).
A: (0, 5) +(5, -1) = (0+5, 5-1) = (5, 4) . . . . . . the points we want
B: (0, 4) +(4, -1) = (4, 3) . . . not it
C: (-1, 5) +(5, 0) = (4, 5) . . . not it
D: (-1, 4) +(4, 0) = (3, 4) . . . not it
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In the attached, the blue line segment is the only one that passes through the midpoint of the given diagonal. It has end point coordinates
(0, 5) and (5, -1) . . . matching choice A
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Answer:
(a) (x - 3)² + (y + 4)² + (z - 5)² = 25
(b) (x - 3)² + (y + 4)² + (z - 5)² = 9
(c) (x - 3)² + (y + 4)² + (z - 5)² = 16
Step-by-step explanation:
The equation of a sphere is given by:
(x - x₀)² + (y - y₀)² + (z - z₀)² = r² ---------------(i)
Where;
(x₀, y₀, z₀) is the center of the sphere
r is the radius of the sphere
Given:
Sphere centered at (3, -4, 5)
=> (x₀, y₀, z₀) = (3, -4, 5)
(a) To get the equation of the sphere when it touches the xy-plane, we do the following:
i. Since the sphere touches the xy-plane, it means the z-component of its centre is 0.
Therefore, we have the sphere now centered at (3, -4, 0).
Using the distance formula, we can get the distance d, between the initial points (3, -4, 5) and the new points (3, -4, 0) as follows;
d = 5
This distance is the radius of the sphere at that point. i.e r = 5
Now substitute this value r = 5 into the general equation of a sphere given in equation (i) above as follows;
(x - 3)² + (y - (-4))² + (z - 5)² = 5²
(x - 3)² + (y + 4)² + (z - 5)² = 25
Therefore, the equation of the sphere when it touches the xy plane is:
(x - 3)² + (y + 4)² + (z - 5)² = 25
(b) To get the equation of the sphere when it touches the yz-plane, we do the following:
i. Since the sphere touches the yz-plane, it means the x-component of its centre is 0.
Therefore, we have the sphere now centered at (0, -4, 5).
Using the distance formula, we can get the distance d, between the initial points (3, -4, 5) and the new points (0, -4, 5) as follows;
d = 3
This distance is the radius of the sphere at that point. i.e r = 3
Now substitute this value r = 3 into the general equation of a sphere given in equation (i) above as follows;
(x - 3)² + (y - (-4))² + (z - 5)² = 3²
(x - 3)² + (y + 4)² + (z - 5)² = 9
Therefore, the equation of the sphere when it touches the yz plane is:
(x - 3)² + (y + 4)² + (z - 5)² = 9
(b) To get the equation of the sphere when it touches the xz-plane, we do the following:
i. Since the sphere touches the xz-plane, it means the y-component of its centre is 0.
Therefore, we have the sphere now centered at (3, 0, 5).
Using the distance formula, we can get the distance d, between the initial points (3, -4, 5) and the new points (3, 0, 5) as follows;
d = 4
This distance is the radius of the sphere at that point. i.e r = 4
Now substitute this value r = 4 into the general equation of a sphere given in equation (i) above as follows;
(x - 3)² + (y - (-4))² + (z - 5)² = 4²
(x - 3)² + (y + 4)² + (z - 5)² = 16
Therefore, the equation of the sphere when it touches the xz plane is:
(x - 3)² + (y + 4)² + (z - 5)² = 16