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In a CAD package, mirror image of a 2D point P(5, 10) is to be obtained about a line which passes through the origin and makes an angle of 450 counterclockwise with the X-axis. The coordinates of the transformed point will be
- a)(7.5, 5)
- b)(10, 5)
- c)(7.5, -5)
- d)(10, -5)
Correct answer is option 'B'. Can you explain this answer?
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In a CAD package, mirror image of a 2D point P(5, 10) is to be obtaine...
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In a CAD package, mirror image of a 2D point P(5, 10) is to be obtaine...
Given information:
- Point P has coordinates (5, 10).
- The mirror image of point P is to be obtained about a line passing through the origin.
- The line makes an angle of 450 counterclockwise with the X-axis.
Approach:
To find the coordinates of the transformed point, we need to understand the concept of mirroring a point about a line.
Explanation:
1. The line passing through the origin and making an angle of 450 counterclockwise with the X-axis divides the coordinate plane into two equal halves.
2. Any point on one side of the line is mirrored to the other side by reflecting it across the line.
3. To find the coordinates of the mirrored point, we need to find the perpendicular distance between the line and point P.
4. The perpendicular distance can be determined using the formula: d = |ax + by + c| / sqrt(a^2 + b^2), where (a, b) is the direction vector of the line and (x, y) are the coordinates of the point.
5. For the given line, the direction vector is (cos 450, sin 450) = (cos 450, -sin 450) = (1/sqrt(2), -1/sqrt(2)).
6. The equation of the line passing through the origin can be written as: x*cos 450 - y*sin 450 = 0.
7. Substituting the coordinates of point P into the equation, we get: 5*(1/sqrt(2)) - 10*(-1/sqrt(2)) = 15/sqrt(2) = 15sqrt(2)/2.
8. The perpendicular distance between the line and point P is 15sqrt(2)/2.
9. To find the coordinates of the mirrored point, we need to move twice the perpendicular distance in the opposite direction.
10. The coordinates of the mirrored point will be: (5 - 2*(15sqrt(2)/2), 10 - 2*(15sqrt(2)/2)) = (5 - 15sqrt(2), 10 - 15sqrt(2)) = (5 - 10.61, 10 - 10.61) = (10, 5).
Therefore, the correct answer is option B: (10, 5).
- Point P has coordinates (5, 10).
- The mirror image of point P is to be obtained about a line passing through the origin.
- The line makes an angle of 450 counterclockwise with the X-axis.
Approach:
To find the coordinates of the transformed point, we need to understand the concept of mirroring a point about a line.
Explanation:
1. The line passing through the origin and making an angle of 450 counterclockwise with the X-axis divides the coordinate plane into two equal halves.
2. Any point on one side of the line is mirrored to the other side by reflecting it across the line.
3. To find the coordinates of the mirrored point, we need to find the perpendicular distance between the line and point P.
4. The perpendicular distance can be determined using the formula: d = |ax + by + c| / sqrt(a^2 + b^2), where (a, b) is the direction vector of the line and (x, y) are the coordinates of the point.
5. For the given line, the direction vector is (cos 450, sin 450) = (cos 450, -sin 450) = (1/sqrt(2), -1/sqrt(2)).
6. The equation of the line passing through the origin can be written as: x*cos 450 - y*sin 450 = 0.
7. Substituting the coordinates of point P into the equation, we get: 5*(1/sqrt(2)) - 10*(-1/sqrt(2)) = 15/sqrt(2) = 15sqrt(2)/2.
8. The perpendicular distance between the line and point P is 15sqrt(2)/2.
9. To find the coordinates of the mirrored point, we need to move twice the perpendicular distance in the opposite direction.
10. The coordinates of the mirrored point will be: (5 - 2*(15sqrt(2)/2), 10 - 2*(15sqrt(2)/2)) = (5 - 15sqrt(2), 10 - 15sqrt(2)) = (5 - 10.61, 10 - 10.61) = (10, 5).
Therefore, the correct answer is option B: (10, 5).
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In a CAD package, mirror image of a 2D point P(5, 10) is to be obtained about a line which passes through the origin and makes an angle of 450 counterclockwise with the X-axis. The coordinates of the transformed point will bea)(7.5, 5)b)(10, 5)c)(7.5, -5)d)(10, -5)Correct answer is option 'B'. Can you explain this answer?
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