To construct a tangent to circle O at point A, follow these steps:
1. Draw circle O with center O and a point A on the circumference.
2. Place the compass on point A and draw an arc that intersects the circle at two points, B and C.
3. Without changing the compass width, place the compass on point B and draw an arc that intersects the previous arc at point D.
4. Draw a line segment from point A to point D. This line is a tangent to circle O at point A.
To construct two tangents to circle O, at points A and B where AB is a diameter, follow these steps:
1. Draw circle O with center O and a diameter AB passing through O.
2. Draw radii from points O to A and O to B.
3. Place the compass on point A and draw an arc that intersects the circle at two points, C and D.
4. Without changing the compass width, place the compass on point B and draw an arc that intersects the circle at two points, E and F.
5. Draw a line segment from point A to point C and from point B to point F. These two lines are tangents to circle O.
The relationship between the two tangents is that they are parallel to each other. This is because when a tangent is drawn to a circle, it forms a right angle with the radius of the circle at the point of tangency. Since AB is a diameter and the radii OA and OB are perpendicular to AB, the tangents drawn at A and B will also be perpendicular to AB. Since perpendicular lines are always parallel to each other, the two tangents will be parallel.
In the case where OA and OB are perpendicular radii, the relationship between the two tangents remains the same. The tangents drawn at points A and B will still be parallel to each other because they are perpendicular to the radii OA and OB, which are perpendicular to each other. So, the tangents will always be parallel in this scenario as well.
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