Coordinate Geometry

Step 1: Define the setup.
Let the center of the circle be , and let be an external point from which two tangents and are drawn to the circle. The points of contact are and . We are asked to prove that and that the angles and are equal.
Step 2: Prove the equality of the tangents.
By the property of tangents to a circle, the tangents drawn from an external point to the circle are always equal in length. That is:
Step 3: Prove the equality of the angles.
In triangles and , we have:
- (radii of the same circle)
- (tangents from the same external point)
- (by the property of tangents)
Thus, by the congruence of the two triangles ( ), we conclude that:
Step 4: Conclusion.
Therefore, the lengths of the tangents are equal, and the angles subtended by the tangents at the center are also equal.

Step 1: Define the setup.
Let the center of the circle be , and let be an external point from which two tangents and are drawn to the circle. The points of contact are and . We are asked to prove that the angle between the two tangents is supplementary to the angle subtended by the line-segment at the center.
Step 2: Use the property of tangents.
We know that the angle between two tangents drawn from an external point to a circle is half of the angle subtended by the line-segment joining the points of contact at the center. That is:
Step 3: Prove the supplementary angle.
Since the total angle around a point is , the angle between the tangents and the angle subtended by the line-segment at the center must add up to . Therefore, we have:
Step 4: Conclusion.
Thus, we have proved that the angle between the two tangents is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

Step 1: Formula for the area of a triangle.
The area of a triangle with vertices , , and is given by:
Step 2: Substitute the coordinates of the vertices.
The coordinates of the vertices are:
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Substitute these values into the area formula: Thus, the area of the triangle is 10.5 square units.

The distance of a point from the origin is given by the distance formula: Substituting the coordinates into the formula: Final Answer: 6.